Graded Rings Corresponding

to Polarised K3 Surfaces

and ℚ-Fano 3 folds

Selma Altınok

Thesis submitted for the degree of Doctor of Philosophy at the University of Warwick

Mathematics Institute University of Warwick

Coventry September 18, 1998

Contents

Acknowledgements iv

Declaration v

Summary vi

Introduction 1

1 Preliminaries 5 1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Basic notions for a nonsingular variety . . . . . . . . . . . . . 6

1.2.1 The Riemann–Roch formula . . . . . . . . . . . . . . . 6 1.2.2 The adjunction formula . . . . . . . . . . . . . . . . . 6

1.3 Basic notions for a singular variety . . . . . . . . . . . . . . . 7 1.3.1 Singularities . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3.2 The Riemann–Roch formula for a singular surface … 10 1.3.3 The Riemann–Roch formula for a ℚ-Fano 3 fold …. 10

1.4 Weighted projective spaces . . . . . . . . . . . . . . . . . . . . 11

2 Numerical Data and the Corresponding Hilbert Function 13 2.1 Numerical data . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 The Riemann–Roch formula on a polarised K3 surface …. 18

3 The Hilbert Function Method 23 3.1 The Hilbert series . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 Searching strategy . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3 Outline of the method . . . . . . . . . . . . . . . . . . . . . . 28 3.4 ℚ-Fano 3 folds . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4 Practical Calculations 33 4.1 Weighted complete intersections of codimension ≤ 2 . . . . . . 33 4.2 Pfaffians in codimension 3 . . . . . . . . . . . . . . . . . . . . 36 4.3 Calculation of plausible K3 surfaces of codimension ≤ 3 for

2 ≤ P_{1} ≤ 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5 Unprojection-Projection 52 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.2 Existence of K3 surfaces in codimension 3 . . . . . . . . . . . 55

5.2.1 List of K3 surfaces in codimension 3 . . . . . . . . . . 68 5.3 Some examples in codimension 4 . . . . . . . . . . . . . . . . 71

5.3.1 Lists of possible K3 surfaces in codimension 4 . . . . . 83

Bibliography 91

List of Tables

3.1 Candidates for ℚ-Fano 3 folds in codimension 4 . . . . . . . . 31

5.1 Weighted K3 surfaces in codimension 3 . . . . . . . . . . . . . 68 5.1 Weighted K3 surfaces in codimension 3 . . . . . . . . . . . . . 69 5.1 Weighted K3 surfaces in codimension 3 . . . . . . . . . . . . . 70 5.1 Weighted K3 surfaces in codimension 3 . . . . . . . . . . . . . 71 5.2 Weighted K3 surfaces in codimension 4 . . . . . . . . . . . . . 83 5.2 Weighted K3 surfaces in codimension 4 . . . . . . . . . . . . . 84 5.2 Weighted K3 surfaces in codimension 4 . . . . . . . . . . . . . 85 5.2 Weighted K3 surfaces in codimension 4 . . . . . . . . . . . . . 86 5.2 Weighted K3 surfaces in codimension 4 . . . . . . . . . . . . . 87 5.2 Weighted K3 surfaces in codimension 4 . . . . . . . . . . . . . 88 5.3 Candidates for weighted K3 surfaces in codimension 4 . . . . . 88 5.3 Candidates for weighted K3 surfaces in codimension 4 . . . . . 89

Acknowledgements

I would first like to thank my supervisor, Prof. Miles Reid, for his help and support during the preparation of this thesis.

This work was supported by the Turkish Higher Educational Council and Aydın Adnan Menderes University, Turkey.

I would like to thank Dr. Mohan Bhupal, Dr. Gavin Brown, Prof. Shigeru Mukai for many useful conversations and also Nagoya University Program for Academic Exchange for their support during my stay in Japan. I would also like to thank Alastair Craw reading part of this thesis.

I would like to end by thanking my family for their support and encouragement throughout the duration of this work.

Declaration

I declare that the work contained in this thesis is original, to the best of my knowledge, except where otherwise stated.

Summary

In this thesis we study polarised varieties (X, D), especially K3 surfaces and ℚ-Fano 3 folds, and the corresponding graded rings R(X, D). The general aim is to describe graded rings explicitly by generators and relations which in good cases can be used to get results for X and its moduli. A polarised variety (X, D) is a projective variety and a polarisation is given by a ℚ-ample Weil divisor (that is, some multiple of D is Cartier and ample). Usually, our varieties are singular and the singularities are canonical singularities. We start with the Hilbert function P_{n} = dim H^{0}(X, nD) given by an appropriate form of the Riemann–Roch formula and write the Hilbert series of P_{n} as a rational function which tells us the possible shape of the ring R(X, D). Then we study varieties and their graded rings by a number of methods such as complete intersection, Pfaffian and projection methods.

We produce lists of families of polarised K3 surfaces (X, D) with a given ℚ-ample Weil divisor and at worst Du Val singularities for which the corresponding graded ring R(X, D) has embedding codimensions 3 and 4, and similarly, ℚ-Fano 3 folds in codimensions 3 and 4 by using the Hilbert function.

Chapter 1 gives basic notation, definitions and results for nonsingular varieties and varieties with canonical singularities and discusses the forms of the Riemann–Roch formula we use later on.

In Chapter 2 we introduce numerical data on a polarised K3 surface. Then we show that numerical data and the corresponding Hilbert function P_{n} can be given by a simple formula by studying the resolution of singularities as in the case of a nonsingular K3 surface.

In Chapter 3 we give a strategy of the search and an outline of the Hilbert function method.

Chapter 4 is an application of Chapter 3. We show that the argument with P_{1} and P_{2} based on 2 ≤ P_{1} ≤ 3 + codim X for codimension ≤ 3 to cut down the whole universe to a few hundred possibilities.

In Chapter 5 we explain a relation between “projection” and the resolution of singularities and how this under some mild conditions leads to the existence of K3 surfaces in codimensions 3 and 4. We work on model examples and give explicit constructions of K3 surfaces by using birational geometry.

The remaining problems that we hope to settle in the near future are the existence results for ℚ-Fano 3 folds including three new candidates with P_{1} = dim H^{0}(−K_{X}) = 0, some K3 surfaces in codimension 4 and how these examples relate to a known structure theory for codimension 4 Gorenstein ideals.

Introduction

The study of graded rings is very important for the classification of surfaces of general type, del Pezzo surfaces, canonical 3 folds and Fano 3 folds etc. The graded ring R associated to a variety X and a divisor D (typically ample) is defined by

R(X, D) = ⨁_{m∈ℤ}H^{0}(X, mD), with X = Proj R(X, D).

There is a two-way correspondence between (X, D) and its graded ring R = R(X, D). In traditional literature, D is replaced by the canonical divisor K to study canonical models. There are still many open problems of classifying surfaces of general type and 3 folds for given numbers K^{2} and p_{g} to which R(X, D) applies.

We mainly study K3 surfaces X with Du Val singularities and a ℚ-ample Weil divisor D embedded in weighted projective spaces, especially the cases where the ring R is “small”. “Small” means that R can conveniently be expressed in terms of generators and relations, such as hypersurfaces, codimension 2 complete intersections, codimension 3 Pfaffians and known models for codimension 4 rings. Studying varieties in weighted projective spaces come naturally in the sense that taking Proj of a graded ring automatically gives a variety embedded in a weighted projective space. Many our varieties are locally toric. This makes easy to work with singularities by using toric methods (see Danilov [6]). We study singularities of X, especially cyclic quotient singularities, in terms of a nonsingular K3 surface S obtained by resolving them. In this case, the resolution on S is an “almost star-shaped” weighted graph which gives rise to a lattice L. One can go beyond that to describe K3 surfaces analytically. Reid conjectured that the primitive sublattice of L can easily be determined in terms of submultiples of D. If the ring R(X, D) is simple such as hypersurfaces he gave a sketch of proof by using Lefschetz theory (see Reid [22]). Before that, Belcastro proved this by using toric descriptions of “famous” 95 hypersurfaces in her Ph.D. thesis (see Belcastro [3]).

Hypersurfaces and codimension 2 complete intersection K3 surfaces were studied by Reid, Iano-Fletcher and independently by Yonemura. Fletcher gave necessary and sufficient conditions that make codimensions 1 and 2 K3 surfaces quasismooth and he also did similar works on canonical 3 folds and ℚ-Fano 3 folds in codimensions 1 and 2. He showed that there is a ℚ-Fano 3 fold X(12, 14) ⊂ ℙ(2, 3, 4, 5, 6, 7) with empty anticanonical linear system |−K| (see Fletcher [9]).

It is very interesting to know a complete structure of Gorenstein ideals I in a regular local ring A and their minimal resolution, in other words, their relations and syzygies of A/I. A ring is Gorenstein if and only if it is Cohen–Macaulay and the canonical divisor of the corresponding variety is trivial. A Gorenstein (Cohen–Macaulay) ideal means that A/I is Gorenstein (Cohen–Macaulay).

The Hilbert–Burch theorem states that any ideal I of codimension 2 in a regular local ring A with n + 1 generators is Cohen–Macaulay if and only if it is the ideal generated by the n × n minors of an n × (n + 1) matrix. In 1890, Hilbert proved this for a polynomial ring and, later on, in 1968, Burch proved it for a regular local ring. Serre showed that if an ideal I has codimension 2, then I is Gorenstein if and only if the corresponding variety V (I) is a complete intersection.

In 1974, Buchsbaum–Eisenbud [4] gave a structure theorem for a Gorenstein ideal of codimension 3 to obtain a complete determinantal description of such an ideal. Their main result for a Gorenstein ideal I of codimension 3 in a regular local ring A is the following: if an ideal I of codimension 3 is Gorenstein then it is the ideal generated by the Pfaffians of the 2m × 2m submatrices obtained by deleting the ith row and the corresponding column of some (2m + 1) × (2m + 1) skew symmetric matrix M of rank 2m (or simply the submaximal Pfaffians of M). Conversely, every Gorenstein ideal of codimension 3 is given by the submaximal Pfaffians of some skew symmetric matrix.

In 1982 there was an attempt done by Kustin–Miller to give a complete structure theorem on codimension 4 Gorenstein ideals. In 1983 they gave a construction of “big Gorenstein ideals from small ones” [15]. The proof is based on constructing a resolution of big Gorenstein ideals. Their main theorem is the following: let (b) = (b_{1}, …, b_{n}) ⊂ (a) = (a_{1}, …, a_{m}) be Gorenstein ideals of codimensions c + 1, c respectively, and ϕ a map of complexes from the minimal resolution 𝔽 of A/(b) to the minimal resolution 𝔼 of A/(a):

𝔽:  0 ⟶ F_{c} ⟶ ⋯ ⟶ F_{1} ⟶^{b} F_{0}
↓^{ϕ_{c}}  ↓^{ϕ_{1}}  ↓^{ϕ_{0}}
𝔼: 0 ⟶ E_{c+1} ⟶^{a^{t}} E_{c} ⟶ ⋯ ⟶ E_{1} ⟶^{a} E_{0}

where the F_{i} and the E_{i} are isomorphic to a direct sum of A such that

ϕ_{0} = id: A ⟶ A, ϕ_{1} : A^{n} ⟶ A^{m}, …, ϕ_{c}: A ⟶ A^{m}

then the ideal generated by the b_{i} and c_{1} + va_{1}, …, c_{m} + va_{m} is a Gorenstein ideal of codimension c + 1 in A[v] where v is an indeterminate and ϕ_{c}^{t} = (c_{1}, …, c_{m}). Common known examples are hypersurface sections of codimension 3 Pfaffians, ideals generated by the (n − 1) × (n − 1) minors of a generic n × n matrix, especially, ℙ^{2} × ℙ^{2} ⊂ ℙ^{8} and ℙ^{1} × ℙ^{1} × ℙ^{1} ⊂ ℙ^{7}, both given by nine equations and sixteen syzygies.

By using the resolution structure of ideals together with the Hilbert function we develop a method, called the Hilbert function method, which extends and improves Fletcher and Reid’s finite tree search method to produce lists of 70 candidates for K3 surfaces in codimension 3 and at least 138 candidates for K3 surfaces in codimension 4. What the method does is a reflection in combinatorics of the Hilbert series P (t) of the following idea of geometry and algebra: to find a regular sequence y_{0}, y_{1}, y_{2} with weights a_{i} which cuts down the dimension of the ring R three times to give a new ring with zero dimension, in other words multiplying P (t) by (1 − t^{a_{1}})(1 − t^{a_{2}})(1 − t^{a_{2}}) will give the Hilbert series of the new ring, and then to find the generators of the new ring. This can be interpreted geometrically by cutting out transversally the cone of the corresponding variety via these hyperplanes (y_{i} = 0).

In 1997–1998, Reid independently stated the theorem of Kustin–Miller as above by using birational geometry, called a new construction of varieties “unprojection”. He replaces a map of complexes by the idea of adjunction (see Remark 5.10). Unprojection is an application of birational geometry to commutative algebra and provides new examples, particularly K3 surfaces in codimensions 3 and 4 (see Chapter 5). We follow Reid’s approach to prove the existence of K3 surface candidates. In here any construction of new varieties we get by lifting from any codimension we call unprojection. This can be done by adding more than one variable. For example, it can be from codimension 1 to 3 or codimension 2 to 4 etc. Therefore we have different types of unprojection. The first type of unprojection corresponds to Kustin-Miller and Reid’s construction, namely starting codimension c Gorenstein ideals to get codimension c + 1 Gorenstein ideals. This construction works for any c. The second type of unprojection is a special technique works from codimension 1 to 3 and the ideal (a) is not Gorenstein. Unprojection proves the existence of almost all candidates in codimension 3 (see Theorem 5.2) and also many in codimension 4. In general, it is almost impossible to check the quasismoothness of a variety (that is, the variety has only at worst cyclic quotient singularities, see Section 1.4 for details) on each strata of weighted projective space by hand. The main point of using unprojection is to get rid of this problem by using Bertini’s theorem in codimensions 1 and 2 to prove the quasismoothness of candidates in codimensions 3 and 4, although it needs special treatment in some cases (see Chapter 5 for examples).

We use similar techniques to study ℚ-Fano 3 folds. The results for a polarised K3 surface can be extended to a ℚ-Fano 3 fold in Chapter 3. I prove that all ℚ-Fano 3 folds Pfaffian in codimension 3 have general hyperplane sections which are K3 surfaces. In case of codimension 4 there are three candidates. Their existence is still an open question.

Chapter 1

Preliminaries

This chapter introduces some notation, basic concepts and results for an algebraic variety of dimension n over an algebraically closed field k of characteristic zero. Standard references for this material are [11], [18] and [19].

1.1 Notation

k algebraically closed field of characteristic zero k^{∗} multiplicative group of k V projective variety of dim n over k S nonsingular surface K(V) function field of V V^{0} nonsingular locus of V ω_{V} sheaf of regular canonical differentials on V^{0} K_{V} canonical divisor corresponding to ω_{V} 𝔸^{n} affine n-space over k ℙ^{n} projective n-space over k μ_{r} cyclic group of rth roots of unity A_{r−1} cyclic Du Val singularity \frac{1}{r}(1, −1) \frac{1}{r}(a, −a, 1) terminal cyclic quotient singularity B basket of cyclic quotient singularities wt(x_{i}) weighted homogeneous degree of x_{i} k[x_{0}, …, x_{n}] polynomial ring ℙ = ℙ(a_{0}, …, a_{n}) weighted projective space of weights a_{0}, …, a_{n} ℙ(x_{0}, …, x_{n}) ℙ with homogeneous coordinates x_{i} of weights a_{i} Cone(X) affine cone over a weighted projective variety X Cone^{∗}(X) punctured affine cone over X P_{n} = dim H^{0}(X, nD) Hilbert function of a graded ring R for each n P (t) = ∑_{n=0}^{∞} P_{n}t^{n} Hilbert series of P_{n} Pf_{i} defining equations of codimension 3 Pfaffians X

1.2 Basic notions for a nonsingular variety

All varieties are nonsingular in this section.

1.2.1 The Riemann–Roch formula

For a divisor D on an algebraic curve C the Riemann–Roch formula takes the form

χ(C, O_{C}(D)) = χ(O_{C}) + deg D.

For a divisor D on an algebraic surface S it takes the form

χ(S, O_{S}(D)) = χ(O_{S}) + \frac{1}{2}D(D − K_{S}).

If S is a K3 surface, that is K_{S} = O_{S} and H^{1}(S, O_{S}) = 0, we have χ(O_{S}) = 2, therefore

χ(S, O_{S}(D)) = 2 + \frac{1}{2}D^{2}.

1.2.2 The adjunction formula

Let Y be a codimension 1 subvariety in an algebraic variety V of dimension n. Then the adjunction formula takes the form

K_{Y} = (K_{V} + Y)_{|Y} .

It follows that for a curve C on a surface S we have the genus formula

2g(C) − 2 = (K_{S} + C)C

and on a K3 surface C^{2} = 2g(C) − 2 and then C^{2} ≥ −2. For an irreducible curve C on a surface S the arithmetic genus p_{a}C is given by

2p_{a}C − 2 = (K_{S} + C)C.

By the adjunction formula

g(C) = p_{a}C.

1.3 Basic notions for a singular variety

We are interested in studying K3 surfaces with at worst Du Val singularities and a ℚ-ample Weil divisor and ℚ-Fano 3 folds. We first define these varieties and then explain the singularities involved.

A K3 surface is a projective surface X with K_{X} = O_{X} and q(X) = H^{1}(X, O_{X}) = 0 which if nonsingular, we denote by S. By a polarised K3 surface (X, D) we mean a K3 surface X having at worst Du Val singularities and a ℚ-ample Weil divisor D.

A Weil divisor D is a ℚ-ample divisor if some multiple of D is Cartier and ample. For a Weil divisor D on a projective variety V we can associate a sheaf O_{V} (D) of rank one, called a divisorial sheaf, given by

Γ(U, O_{V} (D)) = {f ∈ K(V) | div f + D ≥ 0}

for an open subset U in V.

Let V be a projective variety with at worst canonical singularities. Then V is called a ℚ-Fano variety if −K_{V} is ample.

1.3.1 Singularities

Canonical singularities

A projective variety V has at worst canonical (respectively terminal) singularities if it satisfies the following two conditions:

i. there exists an integer r ≥ 1 such that rK_{V} is a Cartier divisor;

ii. if f: W → V is a resolution of V and {E_{i}} the family of exceptional prime divisors, then

rK_{W} = f^{∗}(rK_{V}) + ∑ a_{i}E_{i}

where a_{i} ≥ 0 (respectively a_{i} > 0).

The smallest r for which rK_{V} is Cartier in a neighbourhood of a singularity P ∈ V is called the index of P.

In the case of a surface, the canonical singularities are the nonsingular points together with the Du Val singularities.

Du Val singularities

Du Val singularities can be characterised in many different ways [20].

A surface X has a Du Val singularity P if the following two conditions hold:

i. ω_{X} is invertible at P;

ii. if f: S → X is a minimal resolution of X at P then f^{∗}ω_{X} = ω_{S}.

It follows immediately that a Du Val singularity is a canonical singularity of index 1.

An alternative characterisation is the following: P ∈ X is locally analytically isomorphic to a quotient singularity 𝔸^{2}/G, where G ⊂ SL(2, k) is a finite group. Another is: Du Val singularities are hypersurface singularities with one of the following normal forms

A_{n}: x^{2} + y^{2} + z^{n+1} = 0 for n ≥ 1,
D_{n}: x^{2} + y^{2}z + z^{n−1} = 0 for n ≥ 4,
E_{6} : x^{2} + y^{3} + z^{4} = 0,
E_{7} : x^{2} + y^{3} + yz^{3} = 0,
E_{8} : x^{2} + y^{3} + z^{5} = 0.

For example, consider the group G = ℤ_{2} acting on the coordinates x_{1}, x_{2} of 𝔸^{2} by (x_{1}, x_{2}) ↦ (−x_{1}, −x_{2}). Then X is locally analytically isomorphic to Spec k[x_{1}^{2}, x_{2}^{2}, x_{1}x_{2}], which corresponds to the singularity A_{1} in the above list.

For our purposes, a non cyclic Du Val singularity can usually be replaced by a basket of cyclic quotient singularities. A basket of cyclic quotient singularities, B, introduced by Reid (Section 8, [19]), is a set of cyclic quotient singularities \frac{1}{r}(a, −a) with a coprime to r. These are not points of X but they make the same contribution as the singularities of X to the Riemann– Roch formula. Similarly, any terminal 3-fold singularity can be considered as a basket B of terminal cyclic quotient singularities \frac{1}{r}(a, −a, 1) where a is coprime to r. Later on, we restrict ourself to K3 surfaces with cyclic quotient singularities and ℚ-Fano 3 folds with terminal cyclic quotient singularities.

Cyclic quotient singularities

Let x_{1}, …, x_{n} be affine coordinates on 𝔸^{n} and assign positive integer weights a_{i} to the x_{i}. A singular point P ∈ V is called a cyclic quotient singularity if it has a neighbourhood which is analytically isomorphic to (𝔸^{n}, 0)/μ_{r}, for some r, where the group action μ_{r} × 𝔸^{n} → 𝔸^{n} is given by

ε(x_{1}, …, x_{n}) = (ε^{a_{1}}x_{1}, …, ε^{a_{n}}x_{n}).

We denote such a singularity by

\frac{1}{r}(a_{1}, …, a_{n}).

The following results are taken from Sections 4 and 5 of [19].

Theorem 1.1 A necessary and sufficient condition for a cyclic quotient singularity \frac{1}{r}(a_{1}, …, a_{n}) to be canonical (respectively terminal) is that

\frac{1}{r} ∑_{i} \overline{la_{i}} ≥ 1 (respectively > 1)

for l = 1, …, r − 1 where denotes the smallest residue modulo r.

By the terminal lemma (see Section 5, [19]) terminal cyclic quotient 3-fold singularities are of the form \frac{1}{r}(a, −a, 1) where a is coprime to r, or equivalently, of the form \frac{1}{r}(1, −1, b) where ab ≡ 1 mod r . Similarly, canonical cyclic quotient surface singularities are of the form \frac{1}{r}(a, −a) with a coprime to r, or equivalently, of the form \frac{1}{r}(1, −1). Canonical cyclic quotient surface singularities are A_{r−1}, Du Val singularities. Indeed, the group μ_{r} acts on the affine coordinates x, y of 𝔸^{2} by

ε: \{ \begin{matrix}x ⟼ εx \\ y ⟼ ε^{−1}y \end{matrix}

where ε is an element of μ_{r}.

Let π be the quotient map 𝔸^{n} → 𝔸^{n} /μ_{r}. Then the group μ_{r} acts on π_{∗}O_{𝔸}^{n} and induces a splitting π_{∗}O_{𝔸}^{n} = L_{0} ⊕ ⋯ ⊕ L_{r−1} into r eigensheaves where

L_{i} = {f ∈ π_{∗}O_{𝔸}^{n} | ε(f) = ε^{i}f for all ε ∈ μ_{r}}.

A singularity (V, P) with a Weil divisor D is a cyclic quotient singularity of type _{i}(\frac{1}{r}(a_{1}, …, a_{n})) if (V, P) is locally isomorphic to \frac{1}{r}(a_{1}, …, a_{n}) and O_{V} (D) ≅ L_{i}. In the case of a surface with a Weil divisor D, \frac{1}{r}(a, −a) is of the form \frac{1}{r}(1, −1) and O_{V} (D) ≅ L_{b} where 1 ≤ b ≤ r − 1 and ab ≡ 1 mod r, equivalently, a cyclic quotient singularity of type _{b}(\frac{1}{r}(1, −1)). For convenience we will denote this \frac{1}{r}(a, −a) of type b.

1.3.2 The Riemann–Roch formula for a singular surface

The Riemann–Roch formula for a given surface X with a finite number of Du Val singularities {Q_{i}} and a ℚ-Cartier divisor D reads

χ(X, O_{X}(D)) = χ(X, O_{X}) + \frac{1}{2}(D^{2} − DK_{X}) + ∑_{i} C_{Q_{i}}(D)  (1.1)

where C_{Q_{i}}(D) is a contribution coming from the singularity of O_{X}(D) at Q_{i} and C_{Q_{i}}(D) ∈ ℚ (Section 9, [19]). In the case of a cyclic quotient singularity Q = \frac{1}{r}(a, −a) the contribution is

C_{Q}(D) = −\frac{b(r − b)}{2r}

where ab ≡ 1 mod r.

The Riemann–Roch formula takes the following form

P_{n} = χ(X, O_{X}(nD)) = χ(X, O_{X}) + \frac{1}{2}((nD)^{2} − (nD)K_{X}) + ∑_{i} C_{Q_{i}}(nD) (1.2)

for n ≥ 1 and P_{0} = 1. For every n ≥ 1, the contribution from a singularity Q = \frac{1}{r}(a, −a) can be calculated by

C_{Q}(nD) = −\frac{\overline{nb}(r − \overline{nb})}{2r}

where denotes the smallest residue modulo r.

1.3.3 The Riemann–Roch formula for a ℚ-Fano 3 fold

Let V be a ℚ-Fano 3 fold with at worst canonical singularities Q. We can assume that Q is of the form \frac{1}{r}(1, −1, b), where r is coprime to b, so that the Riemann–Roch formula of V can be written in the following form:

P_{n} = h^{0}(V, O_{V} (−nK_{V})) = \frac{(2n + 1)n(n + 1)}{12}(−K_{V})^{3}

+ (1 + 2n)χ(O_{V}) − ∑_{Q} l(Q, n + 1) (1.3)

for all n ∈ ℕ where

l(Q, n + 1) = ∑_{k=1}^{n} \frac{\overline{bk}(r − \overline{bk})}{2r}

and where denotes the smallest residue modulo r (see Chapter 2 in Fletcher [9]).

1.4 Weighted projective spaces

Our main aim is to study K3 surfaces and ℚ-Fano with singularities, embedded in a weighted projective space, via their graded rings. We give some definitions and simple properties of weighted projective space. For more details, see Fletcher [9], Dolgachev [7].

Let a_{0}, …, a_{n} be positive integers and consider the graded polynomial ring A = k[x_{0}, …, x_{n}], graded by wt(x_{i}) = a_{i}. The weighted projective space associated to the ring A is defined by

ℙ(a_{0}, …, a_{n}) = Proj A.

An alternative construction is the following: with respect to the k^{∗}-action on the affine coordinates of 𝔸^{n+1} given by

λ(x_{0}, …, x_{n}) = (λ^{a_{0}}x_{0}, …, λ^{a_{n}}x_{n})

ℙ(a_{o}, …, a_{n}) is the quotient (𝔸^{n+1} \0)/k^{∗}. The affine piece x_{i} ≠ 0 is isomorphic to the quotient 𝔸^{n} /ℤ_{a_{i}} where the group ℤ_{a_{i}} acts on the affine coordinates {y_{0}, …, ŷ_{i}, …, y_{n}} via

ε: y_{j} ⟼ ε^{a_{j}}y_{j}, for all j ≠ i

where ε is a primitive a_{i}th root of unity. Here one can think of y_{j} as x_{j}/( \sqrt[a_{i}]{x_{i}})^{a_{j}}.

When no confusion can arise, we use either ℙ or ℙ(x_{0}, …, x_{n}) to denote ℙ(a_{0}, …, a_{n}). Let q_{i} be the point (0, …, 1, …, 0) ∈ ℙ where 1 is in the ith position. We call such points vertices of ℙ. The l-plane spanning q_{i_{1}}, …, q_{i_{l}} will be denoted by q_{i_{1}} … q_{i_{l}} and be called an l-stratum. The 1-dimensional strata will also be called edges and the 2-dimensional strata faces.

Definition 1.2 The graded ring R of a polarised K3 surface (X, D) is

R(X, D) = ⨁_{n≥0} H^{0}(X, O_{X}(nD)) with X = Proj R.

Since D is a ℚ-ample divisor, R is finitely generated. One can go backward and forward between R and X = Proj R.

Definition 1.3 Let X be a closed subvariety of ℙ and π: 𝔸^{n+1} \0 → ℙ the map which sends the point with affine coordinates (x_{0}, …, x_{n}) to the point with weighted homogeneous coordinates (x_{0}, …, x_{n}). The punctured affine cone, Cone^{∗}(X), over X is defined to be π^{−1}(X), and the affine cone, Cone(X), over X the closure of Cone^{∗}(X) in 𝔸^{n+1}.

The affine cone of a closed variety X is an algebraic variety in 𝔸^{n+1}. The ideal of Cone(X) is the homogeneous ideal I(X) of X in A. Note that one can think of Cone(X) as Spec R and X either as a quotient X = Cone^{∗}(X)/k^{∗} or Proj R where R = A/I(X).

Definition 1.4 A closed subvariety X of dimension m in ℙ is quasismooth if Cone(X) is smooth of dimension m + 1 outside the vertex 0.

If X is quasismooth the singularities of X come from the k^{∗}-action and are cyclic quotient singularities.

If the ideal I(X) of X is generated by a regular sequence of homogeneous elements of A, then X is called a weighted complete intersection of multidegree (d_{1}, …, d_{l}). A sufficiently general element in the family of all weighted complete intersections of multidegrees (d_{1}, …, d_{l}) is denoted by X(d_{1}, …, d_{l}) in ℙ(a_{0}, …, a_{n}) where l = n − dim X. We keep the same notation when X is not a complete intersection. When I(X) is principal, we call X a weighted hypersurface.

Numerical Data and the Corresponding Hilbert Function

This chapter is preparation for Chapters 3 and 5. Our aim is to study the graded rings of some numerical data which can give rise to polarised K3 surfaces (X, D), via the Riemann–Roch formula. In this chapter, we introduce the numerical data, namely the initial data and the resolution data associated to certain polarised varieties (X, D). We consider the Riemann– Roch formula P_{n} as a function of the numerical data, called the Hilbert function. We also study the resolution of cyclic quotient singularities to find an answer for the existence of possible candidates. This relation is explained in Chapter 5. As a consequence of the resolution, P_{n} on a polarised K3 surface (X, D) can be written in a more efficient way which leads to a simple formula for h^{0}(X, nD) as in the case of a nonsingular K3 surface. This is an alternative and more geometric way of calculating graded ring via linear systems on a nonsingular K3 surface.

2.1 Numerical data

Let (X, D) be a polarised K3 surface and R(X, D) the graded ring of (X, D) (see for definitions in Section 1.3 and 1.4).

Initial data

Definition 2.1 The initial data of a polarised K3 surface (X, D) is D^{2} and its basket of singularities. By formal initial data we mean a number D^{2} ∈ ℚ together with a basket of singularities B (see Section 1.3.1 for the definition). The Hilbert function P_{n} corresponding to the formal initial data {D^{2}, B} is given by (1.2).

Remark 2.2 For our purposes we assume that the type b of singularity \frac{1}{r}(a, −a) in B is ≤ \frac{r}{2}. This is possible since b and r − b have the same effect on P_{n}.

Remark 2.3 It is known (see Theorem III.9.20, [9]) that the number of A_{r−1} singularities on a K3 surface is finite and the bound for this is given by ∑_{r}(r − 1) ≤ 19.

Resolution data

Let (V, D) be a polarised surface with Du Val singularities. Corresponding to the Weil divisor D, we have a divisorial sheaf O_{V} (D) of rank 1 which corresponds to a whole divisor class of D. The Riemann–Roch formula deals with a divisor class, not with a divisor. Therefore we consider a resolution of singularities along a suitable divisor D′ in the divisor class of D. By abuse of notation D′ is called D.

In the case of a surface with Du Val singularities there exists a minimal resolution σ of the singularities of (V, D) for some D ∈ |D| such that σ: (W, σ^{∗}(D)) → (V, D) with σ^{∗}(D) = B + ∑ a_{i}C_{i} where B is the strict transform of D, the C_{i} are irreducible exceptional curves and a_{i} ∈ ℚ. Since the C_{i} are exceptional and K_{W} C_{i} = 0, the C_{i} are (−2)-curves and form the well known A_{n} (n ≥ 1), D_{n} (n ≥ 4), E_{6}, E_{7} or E_{8} configurations (Chapter 3, [2]).

The pull back of a ℚ-Cartier divisor D is defined to be

σ^{∗}(D) := \frac{1}{r}σ^{∗}(rD)

where r is the smallest integer such that rD is Cartier. Note that σ^{∗}(D) is also ℚ-Cartier.

Proposition 2.4 Let P ∈ V be a germ of a cyclic quotient singularity \frac{1}{r}(a, −a) of type b and D a ℚ-ample Weil divisor on V. Then there exists a minimal resolution σ of the singularity along D, for some D ∈ |D|, and a divisor B meeting C_{b} transversely in a neighbourhood of the exceptional curves such that σ^{∗}(D) = B + ∑ a_{i}C_{i}. In other words, the resolution in W locally looks like an “almost star-shaped” graph with a central node B and a weighted chain of length r − 1 of (−2)-curves

[figure: _{\frac{1}{r}(}; r − b; 2(r − b); …; b(r − b); …; b2; b; ); (∗); …; B; …]

Notation For convenience we abbreviate (∗) by

σ^{∗}(D) = B + \frac{1}{r}(r − b, 2(r − b), …, b(r − b), …, b2, b).

Proof. The singularity P is of the form \frac{1}{r}(1, −1), and O_{V} (D) ≅ L_{b} which is generated by x^{b}, y^{r−b} as an O_{V} -module. We denote div x^{b} and div y^{r−b} by L and M respectively, which are in the linear system |D|. From a toric resolution (see the Appendix of [1] and Section 8 of [6]), we have

σ^{∗}(div x^{r}) = rC_{0} + (r − 1, r − 2, …, 1),
σ^{∗}(div y^{r}) = rC_{r} + (1, 2, …, r − 1)

where C_{0} and C_{r} correspond to the pull backs of π(x = 0) and π(y = 0) respectively under the toric resolution, and where π is the quotient map 𝔸^{2} → V. Hence we can write

σ^{∗}L = \frac{b}{r}σ^{∗}(div x^{r}),
σ^{∗}M = \frac{r − b}{r}σ^{∗}(div y^{r}).

The minimum of σ^{∗}L − bC_{0} and σ^{∗}M − (r − b)C_{r} is

\frac{1}{r}(r − b, 2(r − b), …, b(r − b), b(r − b + 1), …, b2, b).

Therefore

σ^{∗}L = B_{1} + \frac{1}{r}(r − b, …, b(r − b), …, b2, b),
σ^{∗}M = B_{2} + \frac{1}{r}(r − b, …, b(r − b), …, b2, b).

Since any principal effective divisor is locally zero against exceptional divisors, it is sufficient to consider either σ^{∗}L or σ^{∗}M. □

Corollary 2.5 We have

BE_{i} = 0 for i ≠ b, otherwise 1,
D^{2} = σ^{∗}(D)σ^{∗}(D) = σ^{∗}(D)B,

D^{2} = B^{2} + \frac{1}{r}b(r − b).

Corollary-Definition 2.6 If there is a basket of cyclic quotient singularities, then

σ^{∗}(D) = B + ∑_{(r,b)} \frac{1}{r}(r − b, 2(r − b), …, b(r − b), …, b2, b),

D^{2} =B^{2} + ∑_{(r,b)} \frac{1}{r}b(r − b), the resolution data,  (2.1)

where B^{2} ∈ 2ℤ and the fractional part of (2.1) corresponds to the contribution of a basket of cyclic quotient singularities to the Riemann–Roch formula and (r, b) represents a cyclic quotient singularity \frac{1}{r}(a, −a) of type b in B.

Proof. The idea from the proof of Proposition 2.4 can be linearly extended to a basket of singularities. □

By formal resolution data we mean a number D^{2} written in the form

D^{2} = B^{2} + ∑_{(r,b)} \frac{1}{r}b(r − b), −4 ≤ B^{2} ∈ 2ℤ  (2.2)

where the sum is taken over a basket B and (r, b) represents a cyclic quotient singularity \frac{1}{r}(a, −a) of type b with r and a coprime. This is a convenient way of encoding the formal initial data, that is a number D^{2} and a basket of singularities.

By looking at the resolution data we can read off necessary information, namely B^{2}, the singularities and the types of the singularities. We give an example from Fletcher’s list [9].

Example 2.7 Let x, y, z, t be the homogeneous coordinates on ℙ(5, 7, 8, 20) with weights 5, 7, 8, 20 respectively. The general weighted quasismooth homogeneous polynomial X: f(x, y, z, t) = 0 of degree 40 in ℙ(5, 7, 8, 20) with D = O_{X}(1) has the resolution data (see Corollary-Definition 2.6)

D^{2} = \frac{40}{5 · 7 · 8 · 20} = −4 + \frac{3}{4} + 2 × \frac{2 · 3}{5} + \frac{6}{7}.

That is, B^{2} = −4 and the singularities are of the following form

\frac{1}{4}(5, 7) = \frac{1}{4}(1, 3)  of type 1 along the q_{2}q_{3} stratum,
2 × \frac{1}{5}(7, 8) = 2 × \frac{1}{5}(2, 3) of type 2 along the q_{0}q_{3} stratum,
\frac{1}{7}(8, 20) = \frac{1}{7}(1, 6)  of type 1 at the vertex q_{1}.

Indeed, since X is quasismooth the only singularities are due to the k^{∗}-action. Therefore the singularities occur only on the edges and at the vertices of ℙ. Note that from the generality assumption of X the coefficients of f could be nonzero constant, 1 for brevity. Therefore the defining equation of X in a calculation is written as

f(x, y, z, t) = x^{8} + y^{5}x + z^{5} + t^{2} + ⋯

where ⋯ refers to all other possible terms. Since x^{8}, z^{5} and t^{2} ∈ f the vertices q_{0}, q_{2}, q_{3} ∉ X. The polynomial f does not contain any power of y so q_{1} ∈ X. Locally at q_{1},

f(x, 1, z, t) = x^{8} + x + z^{5} + t^{2} + ⋯ in 𝔸^{3} /⟨ε⟩

where ε is a primitive 7th root of unity and acts on the affine coordinates x, z, t of 𝔸^{3} by

ε: \{ \begin{matrix}x ⟼ ε^{5}x \\ z ⟼ ε^{8}z = εz \\ t ⟼ ε^{20}t = ε^{6}t. \end{matrix}

The derivative of f with respect to x is nonzero at this point. Hence by the inverse function theorem z, t are local coordinates around q_{1}. This gives rise to a singularity \frac{1}{7}(1, 6), which is A_{6}. Now consider a one dimensional singular stratum q_{i_{1}}q_{i_{2}} of ℙ such that hcf(a_{i_{1}}, a_{i_{2}}) ≠ 1. Along the stratum q_{0}q_{3}, which is the xt line, f is written as

f_{|(y=z=0)} = x^{8} + t^{2} + x^{4}t ⊂ ℙ(5, 20)

There are exactly two intersection points, which correspond to two singularities 2 × \frac{1}{5}(7, 8) of type 2. Along the stratum q_{2}q_{3}, which is the zt line, f is written as

f_{|(x=y=0)} = z^{5} + t^{2} ⊂ ℙ(8, 20).

There are two intersection points which coincide so there is only one singularity \frac{1}{4}(5, 7) of type 1. Therefore X has only the given cyclic quotient singularities.

Similarly, it can be seen that X(8, 9) in ℙ(1, 2, 3, 4, 7) has the resolution data

D^{2} = \frac{8 · 9}{1 · 2 · 3 · 4 · 7} = −2 + 2 × \frac{1}{2} + \frac{3}{7}.

That is, B^{2} = −2 and the singularities are

2 × \frac{1}{2}(1, 1) of type 1 along the stratum q_{1}q_{3},
\frac{1}{7}(3, 4)  of type 2 at the vertex q_{4}.

2.2 The Riemann–Roch formula on a polarised K3 surface

On a polarised K3 surface (X, D) with Du Val singularities, formula (1.2) takes the form

P_{n} = χ(X, O_{X}(nD)) = 2 + \frac{1}{2}(nD)^{2} − ∑_{(r,b)} \frac{\overline{nb}(r − \overline{nb})}{2r}  (2.3)

for n ≥ 1 and P_{0} = 1, where denotes the smallest residue modulo r.

By using resolution data, we write P_{n} in a more efficient form. We divide the problem into two parts. First if (X, D) has no singularities, then D is nef and big (D^{2} > 0) and these conditions, by the general theory of linear systems on a nonsingular K3 surface, imply that h^{0}(nD) = 2 + \frac{1}{2}(nD)^{2} (see Saint-Donat [23]). Second, in the case of a singular K3 surface our aim is to give a simple formula for h^{0}(X, nD) as in the first case.

First note that

H^{0}(S, nσ^{∗}(D)) = H^{0}(X, nD), for every n ∈ ℕ

so we can simplify the notation by replacing σ^{∗}(D) by D. Here σ is a minimal resolution (S, σ^{∗}(D)) → (X, D). Notice that S is a nonsingular K3 surface. By the definition of a divisorial sheaf O_{S}(D) the fractional part of nD does not contribute to the calculation of H^{0}(S, nD). Therefore

H^{0}(X, nD) = H^{0}(S, ⌞nD⌟), for every n ∈ ℕ

where ⌞nD⌟ is the integral part of nD. The integral part of D is defined to be

⌞D⌟ = B + ∑⌞a_{i}⌟C_{i}

where ⌞a_{i}⌟ is the maximal integer not greater than a_{i}.

We now define F_{n} as the sum of the (r − 1)-dimensional array

F_{n} = ∑_{(r,b)} \frac{1}{r}(r − \overline{nb}, 2(r − \overline{nb}), …, \overline{nb}(r − \overline{nb}), \overline{nb}(r − \overline{nb} − 1), …, 2\overline{nb}, \overline{nb}),

where denotes the smallest residue modulo r, and write

D_{n} = nD − F_{n}

where

nD = nB + ∑_{(r,b)} \frac{n}{r}(r − b, 2(r − b), …, b(r − b), …, b2, b).

The following lemma is preparation for Theorem 2.9. It states that F_{n} is in the fixed part of |nD|.

Lemma 2.8 Let (X, D) be as given above. Then

(i) F_{n}C_{i} = \begin{cases}0 & if i ≠ \overline{nb}, \\ −1 & otherwise \end{cases}  and  D_{n}C_{i} = \begin{cases}0 & if i ≠ \overline{nb}, \\ 1 & otherwise; \end{cases}

(ii) F_{n}^{2} = − ∑_{(r,b)} \frac{\overline{nb}(r − \overline{nb})}{r};

(iii) ⌞F_{n}⌟ is in the fixed part of |⌞nD⌟|.

Proof. (i) is obvious since F_{n} is an arithmetic progression and D_{n} = nD −F_{n}. Clearly (i) implies (ii). To prove (iii), it is sufficient to prove that D_{n} is maximal nef against the exceptional divisors C_{i}, that is, D_{n} + H is not nef if H = ∑_{l=1}^{m} b_{i_{l}}C_{i_{l}}, b_{i_{l}} > 0.

Observe that |nD| = |⌞nD⌟|+fractional part and nD = D_{n} + ⌞F_{n}⌟ + (F_{n} − ⌞F_{n}⌟) imply that ⌞nD⌟ = D_{n} + ⌞F_{n}⌟ and F_{n} − ⌞F_{n}⌟ = fractional part. We claim that ⌞F_{n}⌟ is in the fixed part of |⌞nD⌟|. It will be proven by contradiction. We assume that there is only one singularity (the argument can easily be extended to the general case). Suppose that D_{n} + H is nef against C_{i} for each C_{i} ⊂ H. If H contains only one exceptional curve, that is H = b_{j}C_{j} for some b_{j} > 0, then

(D_{n} + H)C_{j} = \begin{cases}HC_{j} < 0 & if j ≠ \overline{nb}, \\ 1 + HC_{j} < 0 & otherwise, \end{cases}

a contradiction.

Now assume that H contains at least two exceptional divisors. Without loss of generality H can be written

H = ∑_{i=1}^{m} b_{i}C_{i}

where b_{i} > 0 for i = 1, m, b_{i} ≥ 0 otherwise, and C_{i}C_{i+1} = 1 for every i = 1, …, m. If H does not contain C\overline{nb}, that is b\overline{nb} = 0, then by assumption D_{n}C_{i} + HC_{i} ≥ 0, which gives H^{2} ≥ 0, a contradiction. So assume that H contains C\overline{nb}. Then

(D_{n} + H)C_{i} = \begin{cases}HC_{i} ≥ 0 & if i ≠ \overline{nb}, \\ 1 + HC_{i} ≥ 0 & otherwise. \end{cases}

Notice that HC\overline{nb} = −1 otherwise H^{2} ≥ 0, a contradiction. When we calculate the intersections of H against the C_{i}, there are cancellations along the diagonals. Indeed,

HC_{1} = −2b_{1} + b_{2} ≥ 0
HC_{2} = b_{1} − 2b_{2} + b_{3} ≥ 0
HC_{3} = b_{2} − 2b_{3} + b_{4} ≥ 0
⋮
HC_{i−2} = b_{i−3} − 2b_{i−2} + b_{i−1} ≥ 0
HC_{i−1} = b_{i−2} − 2b_{i−1} + b_{i} ≥ 0
HC_{i} = b_{i−1} − 2b_{i} + b_{i+1} = −1 for i = \overline{nb}
HC_{i+1} = b_{i} − 2b_{i+1} + b_{i+2} ≥ 0
HC_{i+2} = b_{i+1} − 2b_{i+2} + b_{i+3} ≥ 0
⋮
HC_{m−1} = b_{m−2} − 2b_{m−1} + b_{m} ≥ 0
HC_{m} = b_{m−1} − 2b_{m} ≥ 0.

Add all HC_{i} except i ≠ \overline{nb} and conclude that

0 ≤ ∑_{i≠\overline{nb}} HC_{i} = −b_{1} − (b_{\overline{nb}−1} − 2b\overline{nb} + b_{\overline{nb}+1}) − b_{m},

but since b_{1}, b_{m} > 0 and HC\overline{nb} = −1, ∑_{i≠\overline{nb}} HC_{i} < 0, which is a contradiction. Therefore D_{n} is maximal nef against the exceptional divisors. If we replace H by ⌞F_{n}⌟, then ⌞nD⌟ = D_{n} + ⌞F_{n}⌟ is not nef which means that there exists an exceptional divisor C_{i} for some i in ⌞F_{n}⌟ such that ⌞nD⌟C_{i} is negative. Hence C_{i} is in the fixed part of |⌞nD⌟|. We can repeat the process by replacing ⌞F_{n}⌟ with ⌞F_{n}⌟ − C_{i} and continuing in this way we can prove that ⌞F_{n}⌟ is in the fixed part of |⌞nD⌟|. □

Now we are ready to state the main theorem of this section.

Theorem 2.9 Given (X,D) with cyclic quotient singularities, then for every integer n ≥ 0

(i) h^{0}(X, nD) = 2 + \frac{1}{2}D_{n}^{2};

(ii) h^{0}(X, nD) = h^{0}(S, D_{n}).

Proof. It can easily be seen that (nD)^{2} = D_{n}^{2} − F_{n}^{2}. Inserting this into (2.3), we get

P_{n} = 2 + \frac{1}{2}D_{n}^{2} − \frac{1}{2}F_{n}^{2} + contribution.

On the other hand, \frac{1}{2}F_{n}^{2}, by Lemma 2.8 (ii), is equal to the contribution. This proves (i). The proof of (ii) is divided into two parts corresponding to whether h^{0}(X, nD) ≥ 1 or is equal to 0. First, we remark that if nD = 0, there is nothing to prove. Suppose that h^{0}(S, nD) ≥ 1. Then we can assume that nD is effective. Any element E ∈ |nD| is written as “good”+“bad” where “good” is a sum of exceptionals which by Lemma 2.8 (iii) contains F_{n} and “bad” is a sum of non-exceptionals. This gives a one-to-one correspondence between |nD| and |D_{n}| so (ii) holds. On the other hand h^{0}(S, nD) = 0 implies that h^{0}(S, D_{n}) = 0. To see this, notice that F_{n} being an effective divisor gives h^{0}(S, D_{n}) ≤ h^{0}(S, nD) by using the short exact sequence

0 → O(D_{n}) → O(nD) → O_{F_{n}}(nD) → 0.

□

Corollary 2.10 D_{n}^{2} is even and ≥ −4. In particular, when n = 1, D_{1}^{2} = B^{2} ≥ −4.

Remark 2.11 The resolution data of a polarised K3 surface (X, D) should satisfy B^{2} even and ≥ −4, and ∑(r − 1) ≤ 19 (see Corollary 2.10 and Remark 2.3).

The following question comes naturally:

D_{n}^{2} ≥ 0 if and only if D_{n} is nef.

It is clear that if D_{n} is nef then by the weak form of the Kleiman Criterion D_{n}^{2} ≥ 0 (see Reid [18], Corollary 2.3, p. 116). We point out that this question is not relevant to our study.

Example 2.12 Given the formal initial data {D^{2} = 7/6, A_{1} and A_{2}} it is known that there exists a K3 surface X(7): (f = 0) ⊂ ℙ(1, 1, 2, 3) in Fletcher’s thesis [9]. To see this, we will compute P_{n} by using Theorem 2.9. When we blow up the singularities along D the resolution is

D = B + \frac{1}{2}C_{1} + \frac{2}{3}C_{2} + \frac{1}{3}C_{3}

where BC_{i} = 1 for i = 1, 2 and BC_{3} = 0. So D^{2} = DB = B^{2} + 1/2 + 2/3 implying that B^{2} = 0. Immediately from here, calculate dim H^{0}(X, D) = dim H^{0}(S, B) = 2, which implies that there are only two generators x_{1}, x_{2} of degree 1. Now consider

2D = 2B + C_{1} + C_{2} + fractional part,

so 2B + C_{1} + E_{2} is maximal nef against the exceptionals. Therefore, by Theorem 2.9,

h^{0}(X, 2D) = h^{0}(S, 2B + C_{1} + C_{2}) = 2 + \frac{1}{2}(2B + C_{1} + C_{2})^{2}

which is four but there are already three generators x_{1}^{2}, x_{2}^{2}, x_{1}x_{2}. Therefore we need another generator y of degree 2. Similarly, h^{0}(X, 3D) = 7 gives one more generator z of degree 3 since there are six generators coming from degree 1 and 2, namely S^{3}(x_{1}, x_{2}), the 3rd symmetric tensor power of (x_{1}, x_{2}), and yS^{1}(x_{1}, x_{2}). If n ≥ 4, it can easily be observed that there is no need for a new generator. When n = 7, h^{0}(X, 7D) = 30 and the number of the monomials of degree 7 in variables x_{1}, x_{2}, y, z is 31. Therefore there is a relation f of degree 7. If n ≥ 8, observe that #{monomials of degree n} = #{monomials of degree n − 7} + h^{0}(X, nD), which implies that in further calculations f is the only relation. Then X(7) = Proj k[x_{1}, x_{2}, y, z]/(f) is the weighted hypersurface in ℙ(1, 1, 2, 3), with the resolution data {D^{2} = 0 + 1/2 + 2/3}.

For large n, it is very difficult to make this calculation. A more efficient way is to use the Hilbert function of a graded ring. We will explain this in a further section.

The Hilbert Function Method

The question we are interested in is the following: given a numerical function P_{n}, does there exist a ring R with P_{n} as its Hilbert function? Usually, P_{n} is obtained from the numerical data for a polarised K3 surface (X, D) (or other classes of varieties) and we expect that X = Proj R together with D = O_{X}(1) and conversely R = R(X, D). One way of approaching the question is to consider the Hilbert series of graded rings as a rational function in Theorem 3.1. This is a convenient way to cope with the difficulties of calculating P_{n} by using the generating function given by the Hilbert series. We explain how this approach leads to lists of possible K3 surfaces in Section 3.3. The question of existence will be addressed in Chapter 5.

All the results we get for polarised K3 surfaces can be extended to ℚ-Fano 3 folds. This will be explained in Section 3.4. The question of existence will not be answered in this thesis.

3.1 The Hilbert series

Let the Hilbert series of P_{n} be P (t) = ∑_{n=0}^{∞} P_{n}t^{n} where P_{n} is given by an appropriate form of the Riemann–Roch formula.

Theorem 3.1 Let (X, D) be a K3 surface with Du Val singularities and R = R(X, D) = ⨁_{n≥0} H^{0}(X, nD) with P_{n} = h^{0}(X, nD). Then the Hilbert series of P_{n} is as follows

P (R, t) = \frac{(1 + t)}{(1 − t)} + \frac{1}{2}D^{2} \frac{t(1 + t)}{(1 − t)^{3}} − ∑_{(r,b)} \frac{1}{2r} \frac{∑_{n=1}^{r−1} \overline{bn}(r − \overline{bn})t^{n}}{1 − t^{r}}  (3.1)

where (r, b) represents the singularity \frac{1}{r}(a, −a) in the basket B (see Section 1.3.1 for notation) such that ab ≡ 1 mod r, and denotes the smallest residue modulo r.

Proof. Take the sum of P_{n} over all n ≥ 0, given by (2.3) (see Section 2.2), to obtain

P (R, t) = ∑_{n=0}^{∞} P_{n}t^{n} = 1 + ∑_{n=1}^{∞} (2 + \frac{1}{2}(nD)^{2} − ∑_{(r,b)} \frac{\overline{nb}(r − \overline{nb})}{2r})t^{n}

and rearrange into the following

P (R, t) = 1 + ∑_{n=1}^{∞} 2t^{n} + \frac{1}{2}D^{2} ∑_{n=1}^{∞} n^{2}t^{n} − ∑_{(r,b)} \frac{1}{2r} ∑_{n=1}^{∞} \overline{nb}(r − \overline{nb})t^{n}. (3.2)

In (3.2) the expressions 1 + 2t^{2} + 2t^{3} + ⋯ and (t + 2^{2}t^{2} + 3^{2}t^{3} + ⋯)D^{2}/2 correspond to the first and second terms of (3.1), and for each element of the basket, the function \overline{nb}(r − \overline{nb}) repeats periodically over the intervals [0, r], [r, 2r], … and is zero at the end points, giving the last term of (3.1). □

Remark 3.2 Because D is ℚ-ample, the graded ring R(X, D) of (X, D) is a finitely generated ring, so a quotient ring of the Noetherian graded ring A = k[x_{0}, …, x_{n}] where wt(x_{i}) = a_{i} (see Reid [19], Section 2). Therefore, by Theorem 13.2, Chapter 5 in [16],

P (R, t) = \frac{f(t)}{∏_{i}(1 − t^{a_{i}})}  (3.3)

where f(t) is a polynomial in ℤ[t]. It is known that in the case of a weighted complete intersection X(d_{1}, …, d_{l}) ⊂ ℙ(a_{0}, …, a_{n}), where l = n − dim X,

f(t) = ∏_{i=1}^{l}(1 − t^{d_{i}})

(see Dolgachev [7]).

Example 3.3 The Hilbert series P (R, t) of X(8, 9) in ℙ(1, 2, 3, 4, 7) with D^{2} = −2 + 2 × 1/2 + 2 · 5/7 is

\frac{(1 − t^{8})(1 − t^{9})}{(1 − t)(1 − t^{2})(1 − t^{3})(1 − t^{4})(1 − t^{7})}

= \frac{(1 + t)}{(1 − t)} + \frac{3}{2 · 7} \frac{t(1 + t)}{(1 − t)^{3}} − 2 × \frac{1}{2 · 2} \frac{t}{1 − t^{2}}

− \frac{1}{2 · 7} \frac{10t + 12t^{2} + 6t^{3} + 6t^{4} + 12t^{5} + 10t^{6}}{1 − t^{7}} .

Here the left hand side gives information about the degrees of the defining equations of X and the ambient space in which X is embedded and the right hand side corresponds to the numerical data (see Theorem 3.1).

3.2 Searching strategy

To produce a list of candidates for polarised K3 surfaces (X, D) of given codimension, we consider all possible baskets of singularities on a K3 surface given by a choice of some numerical data (see Chapter 2), the possibilities being a few thousand (see Remark 2.11). We fix a priori codim X, so we are only in R with ≤ codim X + 3 generators. Here by generators of R we mean generators of the graded ring A where R = A/I(X) (see Remark 3.2). For each basket, we have 0 ≤ P_{1} ≤ 3 + codim X and a bound for P_{2} related to P_{1}.

The search is based on the values of P_{1}, where 0 ≤ P_{1} ≤ 7 (and usually ≤ 6, see Section 4.3), and P_{2}. We fix P_{1} and P_{2} and get a bound on ∑_{(r,b)} b (see Lemma 3.7), hence a lower bound on the r. For each list (b_{1}, …, b_{k}) of b we consider all possible lists (r_{1}, …, r_{k}) of r in a lexicographic order. There are several mechanisms which can kill the cases (r_{1}, …, r_{k}) and (r′_{1}, …, r′_{k}) where r′_{i} > r_{i} for some i and r′_{j} ≥ r_{j} for j ≠ i. Each (r_{1}, …, r_{k}) imposes new generators on the ring. The reason is that by Lemma 3.4, each \frac{1}{r}(a, −a) usually implies that R(X, D) has some extra generators in certain degrees. This can be used together with 3+codim X to exclude many cases; especially effective when 2 ≤ P_{1}, they cut the list of possible candidates to quite a small number. Similarly when we increase r to the next possible value r′, Proposition 3.6 tell us that there will be at least one new generator, but an old generator(s) may be lost. In all of our cases we have found that after a certain value of r all of the old generators are retained. This means that there will be no K3 surfaces of small codimension for further iterations. See Section 4.3 where we carry out the search for 2 ≤ P_{1} ≤ 6. Then we have to work case by case on each possible item, to find if (X, D) exists or, if not, to exclude it.

Lemma 3.4 Let (V, D) be a polarised surface having a cyclic quotient singularity \frac{1}{r}(a, −a) with some multiplicity c where a and r are coprime. Then there exist two generators x, y of R(V, D) where wt(x) and wt(y) are congruent to a, −a modulo r. Moreover, depending on whether c = 1 or > 1, either there is at least a generator z of degree r or there are at least two different generators z_{1}, z_{2} where wt(z_{i}) ≡ 0 mod r.

Proof. The singularity of V arises from the ℂ^{∗}-action and hence the ring R has two generators x, y with weights a_{1} and a_{2} which satisfy a_{i} ≡ ±a (mod r) for i = 1, 2 and a_{1} + a_{2} = lr for some l. Moreover, since rD is a Cartier divisor which is locally isomorphic to O_{V}, there exists a further generator z such that wt(z) ≡ 0 (mod r). If c > 1 then the singularities occur at least on a one dimensional stratum. This implies that there are at least two generators whose weights are divisible by r. □

Remark 3.5 Lemma 3.4 is also a consequence of Theorem 3.1.

As we explained above Proposition 3.6 plays a crucial role in terminating the search.

Proposition 3.6 Let {D^{2}, B} and {D′^{2}, B′} be formal initial data for K3 surfaces, where B = {(r_{i}, b_{i})}_{i∈I} and B′ = {(r′_{i}, b_{i})}_{i∈I} such that r_{i} ≤ r′_{i}. Assume that the corresponding numerical functions P_{n} and P′_{n}, given by the Riemann–Roch formula, satisfy P_{1} = P′_{1}. Then

P_{n} ≤ P′_{n} for every n (< for at least one n).

Proof. We assume that B and B′ differ at only term r < r′ where (r, b) ∈ B and (r′, b) ∈ B′. This can easily be extended to general baskets.

The Riemann–Roch formula gives us

P_{n} = 2 + \frac{n^{2}}{2} (D)^{2} − \frac{\overline{nb}(r − \overline{nb})}{2r} + x_{n}

and

P′_{n} = 2 + \frac{n^{2}}{2} (D′)^{2} − \frac{ñb(r′ − ñb)}{2r′} + x_{n}

where , denote the smallest residues modulo r, r′ respectively and x_{n} is the sum of the contributions of the common singularities in B and B′. By using the assumption P_{1} = P′_{1} we can get rid of D^{2} and D′^{2} and some common terms in P_{n} and P′_{n}. This implies that it is sufficient to prove that

Δ_{n}(r, b) ≤ Δ_{n}(r′, b) for every n

where

Δ_{n}(r, b) = \frac{n^{2}b − \overline{nb}}{2} − \frac{(nb − \overline{nb})(nb + \overline{nb})}{2r}  (3.4)

and Δ_{n}(r′, b) is given by replacing r by r′ and by in (3.4). (This notation was first used by Fletcher for different purposes; see Fletcher [9], Definition 3.5.) One can easily check that Δ_{n}(r_{i}, b_{i}) ∈ ℤ, for each singularity (r_{i}, b_{i}) and each n, by using the Riemann–Roch formula or directly by a numerical calculation. We can write

nb = lr + \overline{nb} and nb = l′r′ + ñb.

for some positive integers l, l′. Since r < r′, 0 ≤ l′ ≤ l. Now the proof is divided into the following two cases:

(i) l = l′,

(ii) l > l′.

First notice that

Δ_{n}(r, b) ≤ Δ_{n}(r′, b) ⟺ ll′(r′ − r) + ñb(l − l′ − 1) + \overline{nb}(l − l′ + 1) ≥ 0. (3.5)

Suppose that (i) holds. Since nb = lr + \overline{nb} = lr′ + ñb and r < r′, \overline{nb} ≥ ñb. It follows that by replacing l′ by l on the right hand side of (3.5) we have

Δ_{n}(r, b) ≤ Δ_{n}(r′, b).

Suppose that (ii) holds. Since r′ > r and l > l′, the inequality on the right hand side of (3.5) holds. Thus Δ_{n}(r, b) ≤ Δ_{n}(r′, b). For large n r′ < nb and so l′ > 0. The expression on the right hand side of (3.5) is always greater than zero. Hence Δ_{n}(r, b) < Δ_{n}(r′, b) for some n, as required. □

Lemma 3.7 Given a basket of cyclic quotient singularities (r, b) on a polarised K3 surface (X, D) then

(i) D^{2} = 2P_{1} − 4 + ∑_{(r,b)} c × \frac{b(r − b)}{r} > 0 where c is the number of times

(r, b) appears. Furthermore, 2P_{1} − 4 = B^{2} where B comes from the resolution data in Chapter 2, Corollary 2.6.

(ii) P_{2} = 4P_{1} − 6 + ∑_{(r,b)} cb, where b ≤ \frac{r}{2}.

Proof. (i) follows from the definition of P_{1} and a ℚ-ample Weil divisor D. For (ii) we can assume that b ≤ \frac{r}{2}. This is possible since b and r − b have the same effect on the Riemann–Roch formula. We can get rid of D^{2} in P_{2} by using (i) to obtain

P_{2} = 4P_{1} − 6 + ∑_{(r,b)} cΔ_{2}(r, b)

Since 2b ≤ r, 2b ≡ 0 or 2b mod r. If 2b ≡ 0 mod r then r = 2, otherwise r > 2. In both cases, that is r ≥ 2,

Δ_{2}(r, b) = b.

□

In Lemma 3.7 P_{2} gives most of the information we need, namely the generators of the ring R and the maximum number of singularities which can occur, especially when 2 ≤ P_{1} ≤ 3 + codim X. We explain this point in Section 4.3. One can also get a general formula P_{i} as in the case of P_{2} as long as ib ≤ r for each (r, b) in B.

3.3 Outline of the method

Having fixed singularities and a number D^{2} (or equivalently, a value of P_{1}, see Lemma 3.7 (i)), an outline of the method is as follows:

Step 1. Consider the Hilbert series P (t) of P_{n} as a rational function P (R, t) in Theorem 3.1.

Step 2. Choose suitable module generators y_{1}, y_{2}, y_{3} of wt(y_{i}) = s_{i} so that

P (R/⟨y_{1}, y_{2}, y_{3}⟩, t) = (1 − t^{s_{1}})(1 − t^{s_{2}})(1 − t^{s_{3}})P (R, t)  (3.6)

is a finite polynomial with positive coefficients and of reasonably small degree in A = k[x_{0}, …, x_{n}] of wt(x_{i}) = a_{i}. We call (s_{1}, s_{2}, s_{3}) a “good choice”.

Step 3. Guess the generators by analysing P (R/⟨y_{1}, y_{2}, y_{3}⟩, t) in Step 2 together with Lemma 3.4 and using this as a trial-and-error strategy on the following step:

Step 4. The generators satisfy the condition

∏_{i=0}^{n}(1 − t^{a_{i}})P (R, t) = f(t)  (3.7)

where f(t) is a polynomial with integer coefficients which comes from the resolution structure of the graded ring R and is calculated in Chapter 4 (see also (3.3)).

We now describe the first three steps in more detail.

Step 1 gives information how to choose module generators of R in Step 2 to make a finite polynomial but it does not guarantee a polynomial with positive coefficients.

Remark 3.8 R is a finite k[y_{1}, y_{2}, y_{3}]-module and a Cohen–Macaulay ring of dimension 3 which implies that R/⟨y_{1}, y_{2}, y_{3}⟩ is Cohen–Macaulay of dimension 0, and a vector space over k where {y_{1}, y_{2}, y_{3}} is a regular sequence. Geometrically, it means that C(X) is cut out transversally by three hyperplanes. Hence the P_{n} of the new ring R/⟨y_{1}, y_{2}, y_{3}⟩ are zero for all but a finite number of n.

The remark states only the existence of module generators in Step 2, on the other hand it does not give explicitly how to choose them. We can overcome these difficulties by choosing weights s_{i} sufficiently large, such that some of them are divisible by the indices of the singularities, to kill the periodicity of P_{n}, or alternatively, the denominators of P (R, t). In this way we can always ensure that (3.6) is a finite polynomial with positive coefficients. But the aim is, at the same time, to make the degree of the equation “reasonably small”, which means that we hope that (3.6) is the Hilbert series of an Artinian ring of small codimension.

Step 3 is difficult to automate but by Lemma 3.4 and using a trial-and-error strategy on the steps we can guess what the generators can be as long as the ring is “small”, especially, in the case of codimension ≤ 3.

If we succeed in Step 4 we say that there is a candidate X ⊂ ℙ(a_{0}, …, a_{n}) for a K3 surface. The existence of candidates will be proven in Chapter 5.

3.4 ℚ-Fano 3 folds

We are interested in finding possible candidates for general ℚ-Fano 3 folds V of codimensions 3 and 4 embedded in weighted projective spaces, especially when P_{1} = 0 (see P_{n} = h^{0}(X, −nK_{V}) in (1.3)). The formal initial data is a number (−K_{V})^{3} ∈ ℚ and a basket B of terminal cyclic quotient singularities \frac{1}{r}(1, −1, b) of type b where r is coprime to b. Without loss of generality we can assume that b ≤ \frac{r}{2} because b and r − b have same effect on P_{n}. This is important for computing P_{2}.

By the result of Kawamata [12] on a ℚ-Fano 3 fold, we have

24 = (−K_{V})c_{2}(V) + ∑(r − \frac{1}{r}), (−K_{V})c_{2}(V) > 0.

Therefore the number of singularities \frac{1}{r}(1, −1, b) on a ℚ-Fano 3 fold is bounded, and ∑_{r}(r − 1) ≤ 23. This implies that the computer search will be finite.

Some results on a ℚ-Fano 3 folds

We state without proof some results on a ℚ-Fano 3 fold. The proofs can be done in a similar way as in the case of a polarised K3 surface.

Let V be a ℚ-Fano 3 fold and R(V, −K_{V}) = ⨁_{n≥0} H^{0}(V, −nK_{V}) its ring with P_{n} = h^{0}(X, −nK_{V}), P_{0} = 1. Then

(i) the Hilbert series of P_{n} is of the form

P (R, t) = \frac{(−K_{V})^{3}}{2} \frac{(t + t^{2})}{(1 − t)^{4}} + \frac{(1 + t)}{(1 − t)^{2}} − ∑_{(r,b)} \frac{1}{2r} \frac{∑_{n=1}^{r−1} \overline{bn}(r − \overline{bn})t^{n}}{(1 − t)(1 − t^{r})}

where (r, b) represents the singularity \frac{1}{r}(a, −a, 1) of index r and type b in the basket B such that ab ≡ 1 mod r, and denotes the smallest residue modulo r. Observe that, by replacing (−K_{V})^{3} by D^{2}, the product of P (R, t) and (1 − t) has the same form as the Hilbert series of a polarised K3 surface (X, D) (compare (3.1)).

(ii) Suppose that V has a terminal quotient singularity \frac{1}{r}(a, −a, 1) with multiplicity c, where a and r are coprime. Then there exist four generators x, y, z, t of R(V, −K) where at least three of wt(t), wt(z), wt(y), wt(x) are congruent to 0, a, −a modulo r. This is a consequence of (i) (compare Lemma 3.4). If c > 1 there exist at least two generators whose weights are divisible by r.

(iii) The given formal initial data {(−K)^{3} ∈ ℚ, basket B}, if we increase one of the singularities (r, b) (that is, increase the index r of the singularity under consideration), keeping the type b fixed, then this increases P_{n} (compare Proposition 3.6).

(iv) (−K_{V})^{3} can be expressed in terms of P_{1} and the contributions coming from singularities. Namely,

(−K_{V})^{3} = 2P_{1} − 6 + ∑_{(r,b)} c × \frac{b(r − b)}{r} > 0

where c is the number of singularities (r, b) repeated. In particular, if P_{1} = 0 then

(−K_{V})^{3} = −6 + ∑_{(r,b)} c × \frac{b(r − b)}{r} > 0.

(v) In general,

P_{2} = 5P_{1} − 10 + ∑_{(r,b)} cb

since b ≤ \frac{r}{2}. When P_{1} = 0 we have

P_{2} = −10 + ∑_{(r,b)} cb,

together with

10 ≤ ∑ cb ≤ 12.

If ∑_{(r,b)} cb = 12 then the first singularity is of the form \frac{1}{2}(1, −1, 1) with some multiplicity.

Note that at most six different types of singularities (repetitions allowed) can occur.

In the case of a ℚ-Fano 3 fold V of codim 3, the shape of the resolution of R over A = k[x_{0}, …, x_{6}] with wt(x_{i}) = a_{i} looks like

∏_{i=0}^{6}(1 − t^{a_{i}})P (R, t) = 1 − ∑_{i=1}^{2m+1} t^{d_{i}} + ∑_{i=1}^{2m+1} t^{e_{i}} − t^{s}

where the d_{i} are the degree of the relations and the e_{i} are the degree of the syzygies, and K_{V} = O_{V} (−1) which is K_{V} = O_{V} (s − ∑ a_{i}).

We follow the same strategy for ℚ-Fano 3 folds as for polarised K3 surfaces checking that the method holds Steps 1–4 (see Section 3.3) modified as appropriate. The Maple program written for a polarised K3 surface embedded in weighted projective space (see the Appendix of this chapter) can easily be extended to a ℚ-Fano 3 fold. In my search I proved that there are no Pfaffian ℚ-Fano 3 folds in codim 3 with P_{1} = 0 . When P_{1} = 0, rings are usually “very big”. On the other hand, we do have the following three candidates for ℚ-Fano 3 folds in codim 4.

Table 3.1: Candidates for ℚ-Fano 3 folds in codimension 4

Nb. Candidate for ℚ-Fano 3 fold Formal Initial Data
1 V ⊂ ℙ(2, 2, 3, 3, 4, 4, 5, 5)  −6 + 11 × \frac{1}{2} + \frac{2}{3}
2 V ⊂ ℙ(2, 3, 4, 5, 6, 7, 8, 9)  −6 + 7 × \frac{1}{2} + \frac{2}{3} + \frac{3·5}{8}
3 V ⊂ ℙ(2, 5, 6, 7, 8, 9, 10, 11) −6 + 7 × \frac{1}{2} + \frac{4}{5} + \frac{3·4}{7}

Let f(t) be ∏_{i=0}^{7}(1 − t^{a_{i}})P (R, t) where P (R, t) is the Hilbert series of R over A = [x_{0}, …, x_{7}]. The polynomial f(t) for each candidate in Table 3.1 is the following:

1. f(t) = t^{27} − 3t^{19} − 3t^{18} − 3t^{17} + 2t^{15} + 6t^{14} + 6t^{13} + 2t^{12} − 3t^{10} − 3t^{9}
−3t^{8} + 1
2. f(t) = t^{43} − 2t^{31} − t^{30} − 2t^{29} − 2t^{28} − t^{27} + 2t^{24} + 2t^{23} + 3t^{22} + 3t^{21}
+2t^{20} + 2t^{19} − t^{16} − 2t^{15} − 2t^{14} − t^{13} − 2t^{12} + 1
3. f(t) = t^{57} − t^{41} − t^{40} − 2t^{39} − t^{38} − 2t^{37} − t^{36} − t^{35} + t^{32} + 2t^{31} + 2t^{30}
+3t^{29} + 3t^{28} + 2t^{27} + 2t^{26} + t^{25} − t^{22} − t^{21} − 2t^{20} − t^{19} − 2t^{18}
−t^{17} − t^{16} + 1

From these, we can read off information about degrees of relations and syzygies. For example, in the case of candidate #1 in Table 3.1 we expect that if it exists it should have 9 relations of degrees 8^{3}, 9^{3}, 10^{3} and 16 syzygies of degrees 12^{2}, 13^{6}, 14^{6}, 15^{2}. Candidate #2 looks related to Fletcher’s example X(12, 14) ⊂ ℙ(2, 3, 4, 5, 6, 7) with the singularities \frac{1}{5}(1, −1, 2), 7 × \frac{1}{2}(1, −1, 1), 2 × \frac{1}{3}(1, −1, 1). The existence could possibly be proven by projecting from the line spanned by q_{6} and q_{7} to Fletcher’s example as we will see for K3 surfaces in Chapter 5. We do not study this further in this thesis but we hope to return to it later on.

Practical Calculations

In this chapter we work through the practical calculation of finding the possible candidates (X, D) where codim X ≤ 3, particularly 2 ≤ P_{1} ≤ 6. It is advantageous to have access to a computer that runs Maple (see Appendix 4.3) to check whether the method holds Steps 2–4 in Chapter 3, Section 3.3 for the given choice of (s_{1}, s_{2}, s_{3}).

4.1 Weighted complete intersections of codimension ≤ 2

Let X be a weighted complete intersection. Then it is given by X = Proj R where R = A/I(X) and A = k[x_{0}, …, x_{n}] with wt(x_{i}) = a_{i}. Since X is c. i., then R is Cohen–Macaulay. Therefore pd_{A} R = codim X where the projective dimension of R, pd_{A} R, denotes the minimum of the lengths of projective resolutions of R over A.

If codim X = 1, then I(X) = ⟨f⟩ and the resolution of R is as follows

0 → A ⟶^{f} A → R → 0.

If codim X = 2, then I(X) = ⟨f, g⟩ and by the Hilbert–Burch theorem (see Introduction) the resolution of R has the following form

0 → A ⟶^{(−g,f)} A ⊕ A \underrightarrow{\binom{f}{g}} A → R → 0

(see Eisenbud [8] for more details).

From the resolution of R, we can calculate the Hilbert series and the canonical divisor of X by taking the Hilbert function and Hom( , ω_{ℙ}) of the resolution respectively. In general the canonical divisor of a weighted complete intersection X(d_{1}, …, d_{l}) in ℙ(a_{0}, …, a_{n}) is K_{X} = O_{X}(d_{1} + ⋯ + d_{l} − a_{0} − ⋯ − a_{n}) and the Hilbert series is of the form

P (R, t) = \frac{f(t)}{∏_{i=0}^{n}(1 − t^{a_{i}})}

where

f(t) = ∏_{i=1}^{l}(1 − t^{d_{i}})

(see [7]).

Now we give some examples.

Example 4.1 Let the initial data be

D^{2} = −4 + 5 × 1/2 + 3/4 + 4/5.

Then by Theorem 3.1 P (R, t) is written as the rational function

P (R, t) = \frac{1 + t}{1 − t} + \frac{1}{2 · 20} \frac{t(1 + t)}{(1 − t)^{3}} − 5 × \frac{1}{2 · 2} \frac{t}{1 − t^{2}}

− \frac{1}{2 · 4} \frac{3t + 4t^{2} + 3t^{3}}{1 − t^{4}} − \frac{1}{2 · 5} \frac{4t + 6t^{2} + 6t^{3} + 4t^{4}}{1 − t^{5}} . (4.1)

We can always spot the first nonzero generator(s) and typically a few more in higher degrees by choosing (s_{1}, s_{2}, s_{3}) such that

(1 − t^{s_{1}})(1 − t^{s_{2}})(1 − t^{s_{3}})P (R, t)

is a finite polynomial with positive coefficients . This can be achieved by choosing (s_{1}, s_{2}, s_{3}) sufficiently large and such that the index of every singularity divides one of the s_{i} to kill the denominators of P (R, t) in (4.1). For example, choose (10, 20, 30) then

(1 − t^{10})(1 − t^{20})(1 − t^{30}) ∑ P_{n}t^{n} = 1 + t^{2} + 2t^{4} + t^{5} + 2t^{6} + ⋯ + t^{60},

a finite polynomial with positive coefficients. From the coefficients of this polynomial, we can observe that P_{1} = 0, P_{2} = 1, P_{3} = 0, P_{4} = 2 and P_{5} = 1. It follows that there exist generators x, y, z of degrees 2, 4, 5 respectively. A good choice for (s_{1}, s_{2}, s_{3}) can be (2, 4, 5). The reason is that certainly we need something which is divisible by 4 and 5 to kill the denominators of P (R, t) and also the s_{i} are relatively small integers having this property. Now repeat the same calculation for (2, 4, 5) as we did for the previous choice, we hope to get a polynomial with positive coefficients. In fact,

(1 − t^{2})(1 − t^{4})(1 − t^{5})P (R, t) = 1 + t^{11}.

This looks like the Hilbert series of an Artinian ring with a further generator t of degree 11. Hence a plausible candidate is X ⊂ ℙ(2, 4, 5, 11). To see the structure of its resolution, we multiply the previous equation by (1 − t^{11}) which gives

(1 − t^{2})(1 − t^{4})(1 − t^{5})(1 − t^{11})P (R, t) = 1 − t^{22}.

Therefore the shape of the resolution of R(X, D) over A = k[x, y, z, t] is

0 → A(−22) ⟶^{f} A → R → 0

where f is the defining equation of X in degree 22. The canonical divisor is K_{X} = O_{X}(22 − 2 − 4 − 5 − 11), which is trivial. It can be seen that Cone(X) is smooth by using Bertini’s theorem in the case of a complete intersection as in Example 2.7 (see Fletcher [9] for more details). Therefore this is a K3 surface X(22) in ℙ(2, 4, 5, 11) of codim 1 with given initial data in Fletcher’s list.

Example 4.2 Consider the case D^{2} = −4 + 1/2 + 2/3 + 3/4 + 2 · 3/5 + 8/9. Then it follows from Theorem 3.1 that

P (R, t) = \frac{1 + t}{1 − t} + \frac{1}{2 · 180} \frac{t(1 + t)}{(1 − t)^{3}} − \frac{1}{2 · 2} \frac{t}{1 − t^{2}} − \frac{1}{2 · 3} \frac{2t + 2t^{2}}{1 − t^{3}}

− \frac{1}{2 · 4} \frac{3t + 4t^{2} + 3t^{3}}{1 − t^{4}} − \frac{1}{2 · 5} \frac{6t + 4t^{2} + 4t^{3} + 6t^{4}}{1 − t^{5}}

− \frac{1}{2 · 9} \frac{8t + 14t^{2} + 18t^{3} + 20t^{4} + 20t^{5} + 18t^{6} + 14t^{7} + 8t^{8}}{1 − t^{9}} .

As in the previous example, to detect a few generators we choose (s_{1}, s_{2}, s_{3}) suitably such that

Q(t) = (1 − t^{s_{1}})(1 − t^{s_{2}})(1 − t^{s_{3}})P (R, t)  (4.2)

is a finite polynomial with positive coefficients. For example, choose sufficiently large (20, 30, 72) to kill the denominators of P (R, t) and to guarantee that Q(t) is a polynomial with positive coefficients. It follows that

(1 − t^{20})(1 − t^{30})(1 − t^{72})P (R, t) = 1 + t^{8} + t^{9} + t^{10} + t^{12} + t^{15} + ⋯ + t^{122} is a finite polynomial with positive coefficients. Hence it can easily be seen that P_{i} = 0 for i = 1, …, 7 and P_{8} = P_{9} = P_{10} = P_{12} = P_{15} = 1. In other words, there exist generators x, y, z, t, u of degrees 8, 9, 10, 12, 15 respectively. Since each of the indices of the singularities divide at least one of 8, 9, 10, the degrees of x, y, z, a good choice for (s_{1}, s_{2}, s_{3}) can be (8, 9, 10). Now multiplying P (R, t) by (1 − t^{8})(1 − t^{9})(1 − t^{10}) gives

(1 − t^{8})(1 − t^{9})(1 − t^{10})P (R, t) = 1 + t^{12} + t^{15} + t^{27}.

Hence a plausible candidate is X ⊂ ℙ(8, 9, 10, 12, 15). To see the structure of its resolution, multiply again by (1 − t^{12})(1 − t^{15}) to obtain

(1 − t^{8})(1 − t^{9})(1 − t^{10})(1 − t^{12})(1 − t^{15})P (R, t) = 1 − t^{24} − t^{30} + t^{54}.

This implies that the shape of the resolution of R(X, D) over A = k[x, y, z, t, u] is

0 → A(−54) ⟶^{(−g,f)} A(−24) ⊕ A(−30) \underrightarrow{\binom{f}{g}} A → R → 0

where f and g are the defining equations of X in degrees 30, 24 respectively. The canonical divisor is K_{X} = O_{X}(24 + 30 − 8 − 9 − 10 − 12 − 15), which is trivial. Again it is not hard to check that it is quasismooth. This is a K3 surface X(24, 30) ⊂ ℙ(8, 9, 10, 12, 15) of codim 2, which appears in the list of Fletcher.

4.2 Pfaffians in codimension 3

The generic form for Gorenstein rings of codim 3 is given by Buchsbaum and Eisenbud [4] namely, it is shown that an ideal I of codim 3 in a regular local ring A is Gorenstein if and only if it is the ideal of 2mth order Pfaffians or the submaximal Pfaffians of some (2m + 1) × (2m + 1) skew-symmetric matrix, and also if R = A/I is Gorenstein, then the free resolution (over A) of R is self-dual, which gives us that the resolution of R is written as follows

0 → A ⟶^{P^{t}} F^{∗} ⟶^{M} F ⟶^{P} A

where F is a free A-module of rank 2m + 1, M a (2m + 1) × (2m + 1) skewsymmetric matrix, P a (2m + 1) × 1 matrix and P^{t} denotes the transpose of P.

The generators Pf_{i} of I, called the relations of R, are the 2mth order Pfaffians of M which are given as the Pfaffians^{1} of the 2m × 2m submatrices obtained by omitting the ith row and the ith column of M, and the degree of the relations are the degree of the entries of P. A syzygy is a relation among the relations.

1The Pfaffian of a skew-symmetric matrix is the square root of its determinant. Formulae in the case m = 2 are given in Remark 4.5.

The resolution of a graded ring

In view of the above preamble, we can rewrite the resolution of R in the following form

0 → A(−s) ⟶^{P^{t}} ⨁_{i=1}^{2m+1} A(−e_{i}) ⟶^{M} ⨁_{i=1}^{2m+1} A(−d_{i}) ⟶^{P} A → R → 0  (4.3)

where M is a (2m+1)×(2m+1) skew-symmetric matrix and P is a (2m+1)×1 vector such that the degrees of the entries are given by the degrees d_{i} of the relations, and the degrees of the syzygies are given by the e_{i}. It can be easily observed that s − d_{i} = e_{i}.

The following Lemma 4.3 is important to find out the degrees of the generators and syzygies of the possible candidates if there are any cancellations in f(t).

Lemma 4.3 d_{j} = e_{i} can hold for at most one i and j. In this situation, j = i. Moreover, d_{j} = e_{j} = s/2.

Proof. Suppose that d_{j} = e_{i} holds for j ≠ i. Since M = (m_{ij}) is skewsymmetric and the degree of the entries are given by deg m_{ij} = e_{i} − d_{j}, m_{ij} = −m_{ji} is a unit. Therefore d_{i} = e_{j}. Without loss of generality we can assume that d_{1} ≤ ⋯ ≤ d_{2m+1} and e_{1} ≤ ⋯ ≤ e_{2m+1}. It follows that

d_{i} = e_{j} = d_{j} = e_{i}.

This reduces M to a lower dimensional matrix. Therefore i = j. Since s = d_{j} + e_{j}, d_{j} = e_{j} = s/2.

The Hilbert series

By taking the Hilbert function over the resolution, we can easily deduce that

P (R, t) = \frac{f(t)}{∏_{i=0}^{n}(1 − t^{a_{i}})}

where

f(t) = 1 − ∑_{i=1}^{2m+1} t^{d_{i}} + ∑_{i=1}^{2m+1} t^{e_{i}} − t^{s}.

For m = 2, by Lemma 4.3 either four relations and four syzygies or five relations and five syzygies can appear in f(t). If the first case happens, that is d_{i} = e_{i} for a unique i, we can find the degree of the fifth equation by simply dividing s by 2. There are cases where this occurs, such as #4, #6, #7, … in Table 5.1.

The canonical sheaf

We know that ω_{ℙ} = O_{ℙ}(− ∑ a_{i}) and ω_{V} = Ext^{N−n}(O_{V} , ω_{ℙ}) where ℙ = ℙ(a_{0}, …, a_{N}) and dim V = n. Take Hom( , ω_{ℙ}) of the resolution (4.3). Deduce that ω_{V} = O_{V} (s − ∑ a_{i}). In case of a K3 surface X, the following are equivalent:

(i) ω_{X} ≅ O_{X}.

(ii) s = ∑ a_{i}.

Remark 4.4

(i) Consider the resolution of R for m = 2

0 → A(−s) ⟶^{P^{t}} ⨁_{i=1}^{5} A(−e_{i}) ⟶^{M} ⨁_{i=1}^{5} A(−d_{i}) ⟶^{P} A → R → 0

where M is a 5 × 5 skew-symmetric matrix and P is a vector such that the degrees of the entries are given by the degrees d_{i} of the relations. The rank of M can be 0, 2, 4. If rk M|_{p_{i}} = 0, 4, then either Cone(X) is singular at p_{i} or p_{i} ∉ Cone(X). Therefore we consider only rk M|_{p_{i}} = 2.

(ii) Cone(X) is nonsingular at p_{i} if and only if either p_{i} ∉ Cone(X) or both p_{i} ∈ Cone(X) and there are three linearly independent Pfaffians locally at p_{i}.

(iii) The singularities of X come from the ℂ^{∗}-action if X is quasismooth.

Remark 4.5 Let M = (a_{ij}) be a 5 × 5 skew-symmetric matrix. Then the submaximal Pfaffians, or the 4th Pfaffians, Pf_{i} of M are given by taking the square root of the determinants of the submatrices obtained by deleting the ith row and the ith column of M. These are given by the formulas:

Pf_{1} = a_{23}a_{45} − a_{24}a_{35} + a_{25}a_{34}, Pf_{2} = a_{13}a_{45} − a_{14}a_{35} + a_{15}a_{34},
Pf_{3} = a_{12}a_{45} − a_{14}a_{25} + a_{15}a_{24}, Pf_{4} = a_{12}a_{35} − a_{13}a_{25} + a_{15}a_{23},
Pf_{5} = a_{12}a_{34} − a_{13}a_{24} + a_{14}a_{23}.

Example 4.6 Let the formal initial data be

D^{2} = −4 + \frac{4}{5} + \frac{5}{6} + 2 × \frac{2 · 3}{5}.

Then by Theorem 3.1

P (R, t) = \frac{1 + t}{1 − t} + \frac{1}{2 · 30} \frac{t(1 + t)}{(1 − t)^{3}} − \frac{1}{2 · 5} \frac{4t + 6t^{2} + 6t^{3} + 4t^{4}}{1 − t^{5}}

− \frac{1}{2 · 6} \frac{5t + 8t^{2} + 9t^{3} + 8t^{4} + 5t^{5}}{1 − t^{6}} − 2 × \frac{1}{2 · 5} \frac{6t + 4t^{2} + 4t^{3} + 6t^{4}}{1 − t^{5}} .

We are looking for a good choice for (s_{1}, s_{2}, s_{3}) such that

(1 − t^{s_{1}})(1 − t^{s_{2}})(1 − t^{s_{3}}) ∑ P_{n}t^{n}

is a finite polynomial with positive coefficients. Indeed, some of the s_{i} must be divisible by 6 and 5 to kill the denominators of P (R, t). We also expect two generators whose weights are divisible by 5 since the singularities of type 2, 2 × \frac{1}{5}(3, −3), occur at least on a one dimensional stratum. So (s_{1}, s_{2}, s_{3}) can be (5, 5, 6). After multiplying P (R, t) by (1−t^{5})(1−t^{5})(1−t^{6}) we obtain

(1 − t^{5})(1 − t^{5})(1 − t^{6})P (R, t) = 1 + t^{7} + t^{8} + t^{9} + t^{16}.  (4.4)

This looks like the Hilbert series of an Artinian ring with further generators of degrees 7, 8, 9 so a plausible candidate is X ⊂ ℙ(5, 5, 6, 7, 8, 9). Let x_{1}, x_{2}, y, z, t, w be homogeneous coordinates on ℙ(5, 5, 6, 7, 8, 9) with weights 5, 5, 6, 7, 8, 9 respectively. To see the structure of its resolution of R, we multiply (1 − t^{7})(1 − t^{8})(1 − t^{9}) by (4.4) giving

(1 − t^{5})(1 − t^{5})(1 − t^{6})(1 − t^{7})(1 − t^{8})(1 − t^{9})P (R, t) = 1 − t^{14} − t^{15} − t^{16} − t^{17} − t^{18} + t^{22} + t^{23} + t^{24} + t^{25} + t^{26} − t^{40}.

This means that the shape of the resolution of R over A = k[x_{1}, x_{2}, y, z, t, w] takes the form

0 → A(−40) ⟶^{P^{t}} A(−22) ⊕ A(−23) ⊕ A(−24) ⊕ A(−25) ⊕ A(−26)

⟶^{M} A(−18) ⊕ A(−17) ⊕ A(−16) ⊕ A(−15) ⊕ A(−14) ⟶^{P} A → R → 0.

That is, it is expected that there are five relations in degrees 14, 15, 16, 17, 18 and five syzygies in degrees 22, 23, 24, 25, 26. It is clear that K_{X} is trivial since s = ∑ a_{i} = 40.

The last thing is to check whether X is quasismooth or not. It is enough to examine whether Cone(X) is smooth or not on each stratum of ℙ. Now consider a 5 × 5 skew-symmetric matrix M which comes from the resolution of R so that the degrees of the entries of M are e_{i} − d_{j}. Now we fill in the entries of M with general elements of A as follows:

M = (\begin{matrix}0 & l_{1}(x_{1}, x_{2}) & y & z & t \\ & 0 & z & t & w \\ & & 0 & w & q(x_{1}, x_{2}) \\ & −sym & & 0 & l_{2}(x_{1}, x_{2})y \\ & & & & 0 \end{matrix})

where the monomials l_{i}, q are linear and quadratic respectively. After a change of coordinates we can assume that both l_{1} and l_{2} contain x_{1} and x_{2} and that q = x_{1}x_{2}. Now consider all the possible strata of ℙ. q_{3}, q_{4}, q_{5}: Since some powers of z, t, w appear in Pf_{i} for some i q_{3}, q_{4}, q_{5} ∉ X. q_{2}: The Pfaffians contain no monomial of the form y^{n} for some n so q_{2} ∈ X. Consider the affine piece y = 1. Since rk M|_{q_{2}} = 2 there are locally three Pfaffians in 𝔸^{5} /⟨ε⟩

Pf_{2} = −x_{1} + wt + zq + x_{2},
Pf_{4} = −w + ql_{1} + tz,
Pf_{5} = −t + z^{2} + wl_{1}

which are linearly independent where ε is a primitive 6th root of unity and acts on the affine coordinates x_{1}, x_{2}, z, t, w of 𝔸^{5} via

ε: (x_{1}, x_{2}, z, t, w) ⟼ (ε^{5}x_{1}, ε^{5}x_{2}, εz, ε^{2}t, ε^{3}w)

By the inverse function theorem x_{2}, z are local coordinates around q_{2}, which gives rise to a singularity \frac{1}{6}(5, 7), that is A_{5} of type 1. q_{0}q_{1}: Since y = z = t = w = 0

M|_{q_{0}q_{1}} = (\begin{matrix}0 & l_{1}(x_{1}, x_{2})|_{q_{0}q_{1}} & 0 & 0 & 0 \\ & 0 & 0 & 0 & 0 \\ & & 0 & 0 & q(x_{1}, x_{2})|_{q_{0}q_{1}} \\ & −sym & & 0 & 0 \\ & & & & 0 \end{matrix})

which gives us three intersection points on X ∩ q_{0}q_{1} which are the solutions of l_{1}(x_{1}, x_{2})q(x_{1}, x_{2})|_{q_{0}q_{1}} = 0. These are the points q_{0} = (1, 0, 0, 0, 0, 0), q_{1} = (0, 1, 0, 0, 0, 0) and (a, b, 0, 0, 0, 0) where a and b are nonzero. On q_{0} rk M|_{q_{0}} = 2 and there are three linearly independent equations locally around q_{0}

Pf_{3} = −y + zw − t^{2} + ⋯,
Pf_{4} = −x_{2} + yw − zt + ⋯,
Pf_{5} = −w + yt − z^{2} + x_{2}w.

This means that Cone(X) is smooth at q_{0}. By the inverse function theorem z, t are local coordinates around q_{0}. Under the ℂ^{∗}-action it gives rise to a cyclic quotient singularity \frac{1}{5}(7, 8), that is A_{4} of type 2. On q_{1} rk M|_{q_{1}} = 2. Again write all possible Pfaffians, observe that three of them are linearly independent locally at q_{1}

Pf_{3} = −y + zw − t^{2} + ⋯,
Pf_{4} = −x_{1} + yw − tz + ⋯,
Pf_{5} = −w + yt − z^{2} + · · · .

By the inverse function theorem z, t are local coordinates around q_{1}. This gives a singularity \frac{1}{5}(7, 8) which is A_{4} of type 2. On (a, b, 0, 0, 0, 0) we can again use the change of coordinates x_{1} = l_{1}(x_{1}, x_{2}) and x_{2} = x_{2}. Then our point becomes (0, 1, 0, 0, 0, 0) and M

(\begin{matrix}0 & x_{1} & y & z & t \\ & 0 & z & t & w \\ & & 0 & w & l_{1}(x_{1}, x_{2})x_{2} \\ & −sym & & 0 & l′_{2}(x_{1}, x_{2})y \\ & & & & 0 \end{matrix})

with rk M|_{(0,1,0,0,0,0)} = 2 where l_{2}′ is a linear function in x_{1}, x_{2}. Notice that there are three linearly independent Pfaffians locally around q_{1}

Pf_{1} = −t + w^{2} + zyx_{2} + ⋯,
Pf_{2} = −z + wt + x_{1}y^{2} + ⋯,
Pf_{4} = −x_{1} + yw − zt + · · · .

By the inverse function theorem, w and y are local coordinates at (0,1,0,0,0,0) which give a cyclic quotient singularity \frac{1}{5}(9, 6) of type 1. When we check the quasismoothness candition on the higher dimensional strata the calculations become unmanageable. For this reason we carry out the calculation in the next chapter by using Macaulay (see Example 5.9). This example is #61 in Table 5.1.

Example 4.7 Consider the case:

D^{2} = 2 × \frac{1}{2} + \frac{3}{4}.

It is easy to see that P_{1} = 2 and P_{2} = 5. This gives us at least four generators, say x_{1}, x_{2}, y_{1}, y_{2}, of R in degrees 1, 1, 2, 2. By Lemma 3.4, we must have at least one more generator whose weight is divisible by 4. Therefore a good choice of (s_{1}, s_{2}, s_{3}) can be (1, 1, 4). An alternative way is to consider

P (R, t) = \frac{1 + t}{1 − t} + \frac{7}{2 · 4} \frac{t(1 + t)}{(1 − t)^{3}} − 2 × \frac{1}{2 · 2} \frac{t}{1 − t^{2}} − \frac{1}{2 · 4} \frac{3t + 4t^{2} + 3t^{3}}{1 − t^{4}} .

To kill its denominators, one of the s_{i} must be divisible by 4, say s_{3} = 4. Also, it is already known that there are four generators of weights 1, 1, 2, 2 hence it is enough to choose the rest of the s_{i} from the set of weights 1, 1, 2, 2, for example, s_{1} = s_{2} = 1. After multiplying P (R, t) by (1 − t)(1 − t)(1 − t^{4}) we obtain

P (R/⟨x_{1}, x_{2}, z⟩, t) = 1 + 2t^{2} + t^{3} + 2t^{4} + t^{6}.  (4.5)

This looks like the Hilbert series of an Artinian ring with further generators z, t in degrees 3, 4 and therefore a plausible candidate is X in ℙ(1, 1, 2, 2, 3, 4) that we find below. To find out the structure of the ring R, multiply (4.5) by (1 − t^{2})(1 − t^{2})(1 − t^{3}) to obtain

(1 − t)^{2}(1 − t^{2})^{2}(1 − t^{3})(1 − t^{4})P (R, t) = 1 − t^{4} − 2t^{5} − 2t^{6} + 2t^{7} + 2t^{8} + t^{9} − t^{13}. (4.6)

From here we can read off the shape of the resolution of R(X, D) over A = k[x_{1}, x_{2}, y_{1}, y_{2}, z, t], namely:

0 → A(−13) ⟶^{P^{t}} A(−7) ⊕ A(−7) ⊕ A(−8) ⊕ A(−8) ⊕ A(−9)

⟶^{M} A(−6) ⊕ A(−6) ⊕ A(−5) ⊕ A(−5) ⊕ A(−4) ⟶^{P} A → R → 0.

That is, it is expected that there are five relations in degrees 4, 5, 5, 6, 6 and five syzygies in degrees 7, 7, 8, 8, 9. The canonical divisor is trivial since s = ∑ a_{i} = 13.

We will show that X is quasismooth. In other words, X has only cyclic quotient singularities. It is sufficient to check that Cone(X) is smooth on each strata of ℙ. The equations of X come from 4th order Pfaffians of a skew symmetric M with degrees

(\begin{matrix}0 & 1 & 2 & 2 & 3 \\ 0 & 2 & 2 & 3 \\ 0 & 3 & 4 \\ −sym & 0 & 4 \\ & & 0 \end{matrix}) .

After a change of coordinates we assume that

M = (\begin{matrix}0 & x_{1} & y_{2} & y_{1} & ∗ \\ 0 & 0 & y_{1} + f_{2}(x_{1}, x_{2}) & ∗ & ∗ \\ 0 & z & f_{4}(x_{1}, x_{2}, y_{1}, y_{2}, z) \\ −sym & 0 & t \\ & & 0 \end{matrix})

where f_{2} is a function of x_{1}, x_{2} in degree 2, f_{4} is a function of x_{1}, x_{2}, y_{1}, y_{2}, z in degree 4 and * denotes any possible combinations in the variables x_{1}, x_{2}, y_{1}, y_{2}, z, t in the appropriate degree. q_{0}, q_{1}, q_{2}, q_{3}, q_{4}: Some powers of x_{1}, x_{2}, y_{1}, y_{2}, z appear in Pf_{i} for some i, therefore q_{0}, q_{1}, q_{2}, q_{3}, q_{4} ∉ X. q_{5}: This implies that x_{1} = x_{2} = y_{1} = y_{2} = z = 0. Then

M|_{q_{5}} = (\begin{matrix}0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 \\ −sym & 0 & t \\ & −t & 0 \end{matrix})

with rk M|_{q_{5}} = 2 and so q_{5} ∈ X. Consider the affine piece t = 1. Locally around q_{5}, we have three linearly independent Pfaffians in 𝔸^{5} /⟨ε⟩

Pf_{1} = −x_{1} + f_{5}
Pf_{2} = −y_{1} + h_{6},
Pf_{3} = −y_{2} + g_{6}

where ε is a primitive 4th root of unity, acting on the affine coordinates x_{1}, x_{2}, y_{1}, y_{2}, z of 𝔸^{5} via

ε: (x_{1}, x_{2}, y_{1}, y_{2}, z) ⟼ (εx_{1}, εx_{2}, ε^{2}y_{1}, ε^{2}y_{2}, ε^{3}z),

and f_{5}, h_{6}, g_{6} are the function of x_{1}, x_{2}, y_{1}, y_{2}, z in degrees 5, 6, 6 respectively. In other words Cone(X) is nonsingular along the t-axis. By the inverse function theorem x_{2}, z are local coordinates and under the ℂ^{∗}-action it gives a cyclic quotient singularity \frac{1}{4}(1, 3) which is A_{3}. q_{2}q_{3}q_{5}: This means that x_{1} = x_{2} = z = 0. Then

M|_{q_{2}q_{3}q_{5}} = (\begin{matrix}0 & 0 & y_{2} & y_{1} & 0 \\ 0 & 0 & y_{1} & y_{2} + y_{1} & 0 \\ −y_{2} & −y_{1} & 0 & 0 & y_{1}^{2} + y_{1}y_{2} + y_{2}^{2} \\ −y_{1} & −y_{2} − y_{1} & 0 & 0 & t \\ 0 & 0 & −(y_{1}^{2} + y_{2}^{2} + y_{1}y_{2}) & −t & 0 \end{matrix})

and

rk M|q_{2}q_{3}q_{5} ≤ 2 ⟺ rk (\begin{matrix}−y_{2} & −y_{1} & 0 & 0 & y_{1}^{2} + y_{2}^{2} + y_{1}y_{2} \\ −y_{1} & −y_{2} − y_{1} & 0 & 0 & t \end{matrix}) ≤ 1. (4.7)

The right hand side of (4.7) gives that there are three Pfaffians on q_{2}q_{3}q_{5}

Pf_{1} = −y_{1}t + (y_{1} + y_{2})(y_{1}^{2} + y_{2}^{2} + y_{1}y_{2}),
Pf_{2} = −y_{2}t + y_{1}(y_{1}^{2} + y_{2}^{2} + y_{1}y_{2}),
Pf_{5} = y_{2}(y_{1} + y_{2}) − y_{1}^{2}.

It follows that X ∩ q_{2}q_{3}q_{5} consists of 2 points and that the rank of M at each of these points is two. Since y_{1}, y_{2} and t are non zero at these points, it can be observed that locally

Pf_{2} = −y_{2}t + y_{1}(y_{1}^{2} + y_{2}^{2} + y_{1}y_{2}) + ⋯,
Pf_{3} = −x_{1}t + g_{4}(x_{1}, x_{2}, y_{1}, y_{2}, z),
Pf_{5} = −x_{1}z + y_{2}(y_{1} + y_{2}) − y_{1}^{2} + ⋯,

are linearly independent. Since these points are in the affine piece (y_{1} = 1), by the inverse function theorem, x_{1}, z are local coordinates and hence these give rise to 2A_{1}. It is difficult to check the quasismoothness condition on all of the other higher dimensional strata. This difficulty can be overcome by using unprojection, which will be defined in the next chapter. This example is #13 in Table 5.1.

4.3 Calculation of plausible K3 surfaces of codimension ≤ 3 for 2 ≤ P_{1} ≤ 6

Under the assumption of Lemma 3.7, we have

P_{2} = 4P_{1} − 6 + ∑_{(r_{i},b_{i})} c_{i}b_{i}

where b_{i} corresponds to the type of the singularity r_{i} and c_{i} is the number of times singularity (r_{i}, b_{i}) appears.

Recall that X(d_{1}, …, d_{l}) ⊂ ℙ(a_{0}, …, a_{n}) denotes a sufficiently general element in the family of all weighted varieties given by the zero locus of f_{1}, …, f_{l} where deg f_{i} = d_{i}.

Let P_{1} = 6. Then there exists a complete intersection K3 surface X(2, 2, 2) in ℙ(1, 1, 1, 1, 1, 1) by Bertini’s theorem.

Suppose that P_{1} = 5. So there is at most one singularity. If there are no singularities then there is a c. i. K3 surface X(2, 3) of codim 2 in ℙ(1, 1, 1, 1, 1). If there is a singularity then by Lemma 3.4 it is of the form \frac{1}{2}(1, −1). Clearly X(2, 3, 3, 3, 3) in ℙ(1, 1, 1, 1, 1, 2) with A_{1} is a candidate for a K3 surface (see the Application in the appendix of this chapter).

Suppose that P_{1} = 4. So 10 ≤ P_{2} ≤ 12, that is 0 ≤ ∑ c_{i}b_{i} ≤ 2. Hence there are at most two singularities. Suppose that there are no singularities, that is P_{2} = 10, which gives us a K3 surface X(4) of codim 1 in ℙ(1, 1, 1, 1). Suppose that there is only one singularity, that is P_{2} = 11. Then we have:

r = 2 ⟹ there is a c. i. K3 surface X(3, 3) in ℙ(1, 1, 1, 1, 2) with A_{1},
r = 3 ⟹ X(3, 3, 4, 4, 4) in ℙ(1, 1, 1, 1, 2, 3) with A_{2} is a candidate for

a K3 surface.

When r ≥ 4 we have six generators coming from the case r = 3. In addition we have at least one more generator by either Proposition 3.6 or Lemma 3.4. It follows that there are no K3 surfaces of codim 3 for r ≥ 4. Suppose that there are two singularities. Then P_{2} = 12. It follows that r_{1} = r_{2} = 2. Hence X(3, 3, 3, 3, 4) in ℙ(1, 1, 1, 1, 2, 2) with 2A_{1} is a candidate for a K3 surface.

Let P_{1} = 3. So,

6 ≤ P_{2} = 6 + ∑_{i} c_{i}b_{i} ≤ 9 ⟺ 0 ≤ ∑_{i} c_{i}b_{i} ≤ 3

which implies that there are at most two different kinds of singularities that can appear in the following forms:

c × \frac{1}{r}(1, −1) or \{\frac{1}{r_{1}} (1, −1), \frac{1}{r_{2}} (1, −1)\}

where c ≤ 3 and r_{1} ≠ r_{2}. If P_{2} = 9, then X(3, 3, 4, 4, 4) in ℙ(1, 1, 1, 2, 2, 2) with 3A_{1} is a plausible candidate for a K3 surface. Let P_{2} = 8 and consider, first, only the first possibility 2 × \frac{1}{r}(1, −1). This gives us

r = 2 ⟹ there is a K3 surface X(3, 4) in ℙ(1, 1, 1, 2, 2) with 2A_{1}.

When r ≥ 3, by Lemma 3.4 there are at least two more generators whose weights are divisible by r. Hence the minimum number of generators is greater than six and so the corresponding case will not be considered here. Next, assume the second possibility, that is, there are two singularities of different indices. By Lemma 3.4 r_{1} = 2 and r_{2} = 3 which implies that X(3, 4, 4, 4, 5) in ℙ(1, 1, 1, 2, 2, 3) with A_{1}, A_{2} is a candidate for a K3 surface. Let P_{2} = 7. Then we have only one singularity which is of the form \frac{1}{r}(1, −1). Now we proceed with the algorithm:

r = 2 ⟹ there is a K3 surface X(5) in ℙ(1, 1, 1, 2) with A_{1},
r = 3 ⟹ there is a K3 surface X(4, 4) in ℙ(1, 1, 1, 2, 3) with A_{2},
r = 4 ⟹ X(4, 4, 5, 5, 6) in ℙ(1, 1, 1, 2, 3, 4) with A_{3} is a candidate for

a K3 surface.

When r ≥ 5 we have six generators coming from the case r = 4 plus at least one new generator by Proposition 3.6. Therefore there are no K3 surfaces of codim 3 for r ≥ 5. We can obtain the same result by using Lemma 3.4. Let P_{2} = 6, then there are no singularities. However there is a nonsingular K3 surface X(6) of codim 1 in ℙ(1, 1, 1, 3).

Assume that P_{1} = 2. It is easily seen that 3 ≤ P_{2} ≤ 7. Let P_{2} = 7, then X(4, 4, 4, 4, 4) in ℙ(1, 1, 2, 2, 2, 2) with 5A_{1} is a candidate for a K3 surface. Let P_{2} = 6, then the generators of R with weights {1, 1, 2, 2, 2} imply that at most two kinds of singularities can occur. If cb = 4, then by Lemma 3.4 b = 1. Now we start the algorithm:

r = 2 ⟹ there is a K3 surface X(4, 4) in ℙ(1, 1, 2, 2, 2) with 4A_{1}

and when r ≥ 3 by Lemma 3.4 there are at least two more generators divisible by r so that codim(X) ≥ 4 which is not considered. If c_{1}b_{1} + c_{2}b_{2} = 4, then there are two kinds of singularities of different indices r_{1}, r_{2}. These should be 3A_{1} and A_{2} by Lemma 3.4 and hence X(4, 4, 4, 5, 5) in ℙ(1, 1, 2, 2, 2, 3) is a candidate for a K3 surface with these singularities. Let P_{2} = 5. If cb = 3, then the following possibilities can occur:

(a) r ≥ 2 and b = 1,

(b) r ≥ 7 and b = 3.

Suppose that (a) holds. Then we have

r = 2 ⟹ there is a K3 surface X(6) in ℙ(1, 1, 2, 2) with 3A_{1}

and when r ≥ 3 we have P_{3} = 11, not a case of codim 3 as before. Assume that (b) holds. Then P_{3} ≥ 9 and so by Lemma 3.4 r should be 5 which contradicts r ≥ 7. If c_{1}b_{1} + c_{2}b_{2} = 3, then we have the following cases:

(i) c_{1} = 2 and b_{1}, c_{2}, b_{2} = 1,

(ii) c_{2} = 2 and c_{1}, b_{1}, b_{2} = 1,

(iii) b_{1} = 2 and c_{1}, c_{2}, b_{2} = 1,

(iv) b_{2} = 2 and c_{1}, b_{1}, c_{2} = 1.

Suppose that (i) holds. If r_{1} ≠ 2, then P_{3} = 11 which gives more than seven generators. So consider r_{1} = 2 and increase r_{2} inductively, then

r_{2} = 3 ⟹ there is a K3 surface X(4, 5) in ℙ(1, 1, 2, 2, 3) with 2A_{1}, A_{2},
r_{2} = 4 ⟹ X(4, 5, 5, 6, 6) in ℙ(1, 1, 2, 2, 3, 4) with 2A_{1}, A_{3} is a candidate

for a K3 surface

and when r_{2} ≥ 5 there are no K3 surfaces X of codim 3 by arguing as before. Suppose that (ii) holds. This means that a singularity of index r_{2} occurs on a one dimensional stratum. Therefore there are at least two new generators whose weights are divisible by r_{2}. By Lemma 3.4 r_{2} = 3 which implies r_{1} = 2 and so X(4, 4, 5, 5, 6) in ℙ(1, 1, 2, 2, 3, 3) with A_{1}, 2A_{2} is a candidate for a K3 surface. Assume that (iii) holds. So r_{1} ≥ 5. By Lemma 3.4 we need one more generator whose weight is congruent to 3, but P_{3} = 10. This implies that there are two more generators of degree 3. Hence codim X ≥ 5. Suppose that (iv) holds. Then r_{2} ≥ 5 and so P_{3} ≥ 9, but the only possibilities of r_{1} and r_{2} are 2 and 5 respectively, otherwise P_{3} ≥ 10 when r_{2} ≥ 7.

Therefore X(4, 5, 6, 6, 7) in ℙ(1, 1, 2, 2, 3, 5) with A_{1}, A_{4} is a candidate for a K3 surface. If c_{1}b_{1} + c_{2}b_{2} + c_{3}b_{3} = 3, then P_{3} ≥ 10 which gives at least two more generators of weight 3 but we need one more generator whose weight is r_{3}. Therefore codim(X) ≥ 4. Let P_{2} = 4. This implies that at most two kinds of singularities can occur. Suppose that cb = 2. Then

(a) c = 2 and we have:

r = 2 ⟹ there is a K3 surface X(8) in ℙ(1, 1, 2, 4) with 2A_{1},
r = 3 ⟹ there is a K3 surface X(4, 6) in ℙ(1, 1, 2, 3, 3) with 2A_{2}.

When r ≥ 4 there are no K3 surfaces X of codim 3 by arguing as before, using either Proposition 3.6 or Lemma 3.4.

(b) b = 2 which implies r ≥ 5, therefore

r = 5 ⟹ there is a K3 surface X(6, 6) in ℙ(1, 1, 2, 3, 5) with A_{4}.

When r ≥ 7, P_{3} = 8. This means that we have at least five generators of weights ≤ 3. By Lemma 3.4 we need two more new generators. Hence there are no K3 surfaces X of codim 3 for r ≥ 7.

If c_{1}b_{1} + c_{2}b_{2} = 2, then r_{1} ≠ r_{2}. We are going to consider the following two cases:

(i) r_{1} ≠ 2,

(ii) r_{1} = 2.

Suppose that (i) holds. Then P_{3} = 8 which implies that there are two new generators of degree 3 and hence by Lemma 3.4 r_{1} = 3 and r_{2} = 4. This gives a candidate X(4, 5, 6, 6, 7) in ℙ(1, 1, 2, 3, 3, 4) with A_{2}, A_{3} for a K3 surface. If either r_{1} or r_{2} is ≥ 4, then codim(X) ≥ 4 by arguing as before. Now suppose that (ii) holds. Then

r_{2} = 3 ⟹ there is a K3 surface X(7) in ℙ(1, 1, 2, 3) with A_{1}, A_{2},
r_{2} = 4 ⟹ there is a K3 surface X(5, 6) in ℙ(1, 1, 2, 3, 4) with A_{1}, A_{3},
r_{2} = 5 ⟹ X(6, 6, 7, 7, 8) in ℙ(1, 1, 2, 3, 4, 5) with A_{1}, A_{4} is a candidate

for a K3 surface.

If r_{2} ≥ 6, then there are no K3 surfaces X of codim 3 by arguing as before. Let P_{2} = 3. This implies that there is only one singularity of type 1 such that

r = 2 ⟹ there is a K3 surface X(12) in ℙ(1, 1, 4, 6) with A_{1},
r = 3 ⟹ there is a K3 surface X(10) in ℙ(1, 1, 3, 5) with A_{2},
r = 4 ⟹ there is a K3 surface X(9) in ℙ(1, 1, 3, 4) with A_{3},
r = 5 ⟹ there is a K3 surface X(6, 8) in ℙ(1, 1, 3, 4, 5) with A_{4},
r = 6 ⟹ X(6, 7, 8, 9, 10) in ℙ(1, 1, 3, 4, 5, 6) with A_{5} is a candidate

for a K3 surface.

If r ≥ 7, then there are no K3 surfaces X of codim 3.

So far when 2 ≤ P_{1} ≤ 6, we have fourteen plausible K3 surfaces of codim 3 with the given singularities plus a c. i. nonsingular K3 surface of codim 3.

Appendix

Maple program

The following program, written in the Maple V Release 4 language, is essentially a prearranged group of commands. The reference [17] for Maple V explains how to use Maple V as a programming language.

#Final() should be considered as
#Final(k,D^2,r_1,..,r_k,b_1,..,b_k,a_0,...,a_l)
#=\prod_{i=0}^{l}(1-t^{a_i})P(t) where k is the number of singularities,
#the r_j are the indices of the singularities, the b_j are the types of
#the singularities, the a_i are the weights of either the ring generators
#or the module generators and P(t)=\sum P_nt^n is the Hilbert series
#of P_n for the given initial data, considered as a rational function
#using Theorem 3.1.
Final := proc()
      local sum1, sum2, sum3, sum4, j;
      sum1 := Mult(args[j] $j=(3+2*args[1])..nargs);
      sum2 := simplify(expand(Cpoly(args[j] $j=2..(2+2*args[1]))*sum1));
      sum3 := simplify(simplify(expand(sum1*(1+t))/(1-t))
              +simplify(expand(1/2*args[2]*sum1*t*(1+t))/((1-t)^3)));
      sum4 := sort(simplify(expand(sum2+sum3)));
      sum1*P(t) = sum4;
end:
#Mult(): A procedure for calculating the product of (1-t^{a_i}) over i.
Mult := proc()

     local s, i;
     s := 1;
     for i from 1 by 1 to nargs do
        if args[i]>0 then
        s := (1-t^args[i])*s;
        fi;
     od;
     s;
end:
#Cpoly(): A procedure for rewriting the power series of the sum of the
#contributions of the singularities as a finite sum of some polynomials.
Cpoly := proc()
      local k, l, n, h, j, sum1, sum2, sum3;
      k := (nargs-1)/2;
      for h from 1 by 1 to k do
          sum1[h] := 0 ;
      od;
      sum2 := 0;
      h := 1;
      for l from 2 by 1 to k+1 do
          if args[l]-1>0 then
             for n from 1 by 1 to args[l]-1 do
                 sum1[h] := sum1[h]+Contr(args[l],n*args[k+h+1])*t^n;
             od;
          fi;
          sum2 := sum2 - simplify(1/(2*args[l])*sum1[h]/(1-t^args[l]));
          h := h+1;
      od;
      sum2;
end:
#The following procedure computes twice the contribution C_n(Q) of
#a singularity Q of index r and type b for each n.
Contr := proc(r,nb)
      local result, j;
      j := nb mod r;
      result := j*(r-j);
      result;
end:
#Save: Record the state of all variables and procedures
#in filename.m.
save filename.m

Application

We start Maple sessions with the command “read filename.m”. This is followed by running the procedure Final.

> maple -x
> read filename.m

The following calculation shows how the structure of the resolution of a candidate for a codimension 3 K3 surface X with the singularity A_{1} of type 1 and D^{2} = 13/2 in ℙ(1, 1, 1, 1, 1, 2) looks like via the procedure Final:

> Final(1,13/2,2,1,1,1,2);
         2       2          4    3    2
 (1 - t )  (1 - t ) P(t) = t + 3t + 5t + 3t + 1
> Final(1,13/2,2,1,1,1,2,1,1,1);
        5      2            7    5     4     3   2
(1 - t ) (1 - t ) P(t) = - t  + t  + 4t  - 4t - t + 1

This implies that a candidate for a K3 surface with given data is defined by the common zero locus of five equations of degrees 2, 3, 3, 3, 3. It has also five syzygies of degrees 4, 4, 4, 4, 5.

The following example computes the structure of the resolution of a codimension 3 possible K3 surface with singularities 3A_{1}, A_{6}, A_{6} of types 1, 1, 1, 1, 3 and D^{2} = 1/14 in ℙ(2, 5, 6, 7, 7, 8):

> Final(5,1/14,2,2,2,7,7,1,1,1,1,3,2,7,7);
         7 2       2          16    11    10    8    6    5
   (1 - t )  (1 - t ) P(t) = t   + t   + t   + t  + t  + t  + 1
> Final(5,1/14,2,2,2,7,7,1,1,1,1,3,2,7,7,5,6,8);
      8        6        5        7 2       2           35    23
(1 - t ) (1 - t ) (1 - t ) (1 - t )  (1 - t ) P(t) = -t   + t
        22    21    20    19    16    15    14    13    12
     + t   + t   + t   + t   - t   - t   - t   - t   - t   + 1

In other words, there are five relations in degrees 12, 13, 14, 15, 16 and five syzygies in degrees 19, 20, 21, 22, 23.

Unprojection-Projection

In this chapter, we construct examples of K3 surfaces of codimensions 3 and 4 from smaller codimension K3 surfaces whose rings are much simpler. This can be done in two ways: one way is to use birational geometry and the other way is to use commutative algebra. In all cases except Example 5.14 in codim 4 we explain how to use birational geometry to get quasismooth K3 surfaces. In Example 5.14 we construct the surface (X, D) through an (X, 2D)-model.

5.1 Introduction

Unprojection is an inverse operation of projection. What we mean here by projection is the following: we have a K3 surface X with A_{n} type singularities such that the resolution of these singularities forms a configuration of curves on a nonsingular K3 surface S (as explained in Chapter 2) then contracting some suitable subconfiguration on S with some assumptions gives another K3 surface Y of smaller codimension with A_{n} type singularities, containing some curves C_{i}. The inverse operation is to construct a K3 surface Y with A_{n} singularities, containing the curves C_{i} and use this to get a K3 surface X of higher codimension. Going forward and backward between Y and X is a birational operation, called unprojection-projection.

Now we demonstrate the simplest example of unprojection-projection between K3 surfaces of codimensions 2 and 1. In the first part of the example we consider projection, in the second part we consider unprojection.

Example 5.1 (A) We start by assuming that a K3 surface X = X(3, 3) ⊂ ℙ(1, 1, 1, 1, 2) with A_{1} and a ℚ-ample Weil divisor D = O_{X}(1), where D^{2} = 9/2, exists then we want to construct a nonsingular K3 surface Y = X(4) ⊂ ℙ^{3} with an ample divisor D′, containing the line C: x_{1} = x_{2} = 0. This can be done in two ways: the first way is to blow up the singularity along D and contract down a suitable subconfiguration as explained in the above preamble. The second way is the following: pick the point (0, 0, 0, 0, 1), which is an A_{1} singularity, and do projection away from this point to get Y = X(4) as desired.

We now give details of the two constructions. Let x_{1}, …, x_{4}, y be homogeneous coordinates on ℙ(1, 1, 1, 1, 2) with weights 1, 1, 1, 1, 2.

(1) Using the same notation as in Chapter 2 the resolution of the singular point along D is σ: (S, σ^{∗}(D)) → (X, D) such that σ^{∗}(D) = B + \frac{1}{2}C_{1} where B^{2} = 4 and BC_{1} = 1 (see Proposition 2.4). Now choose the subconfiguration B and consider the corresponding map ϕ_{|B|} from S to ℙ^{3}. If the linear system |B| is not hyperelliptic then the image Y = ϕ_{|B|}(X) ⊂ ℙ^{3} is a K3 surface of degree 4 (see Saint-Donat [23]) with the divisor D′ = (ϕ_{|B|})_{∗}(B), and contains a line C = ϕ_{|B|}(C_{1}) since BC_{1} = 1. In this case it is easy to choose coordinates such that the equations of C are chosen among the coordinates. However we use a construction which is applicable in more general situations.

Consider the following exact sequence:

0 → O_{Y} (−C) → O_{Y} → O_{C} → 0.  (5.1)

Twisting by D′ (in general we may need to twist (5.1) by nD′ where n > 1) and then taking cohomology, we have

0 → H^{0}(Y, O_{Y} (D′ − C)) → H^{0}(Y, O_{Y} (D′)) → H^{0}(C, O_{C}(D′)) → ….

Since H^{0}(Y, O_{Y} (D′)) = H^{0}(X, O_{X}(B)) the middle vector space has dimension 4, and the right hand vector space has dimension 2 by Riemann– Roch on C_{1}. So we conclude that

dim H^{0}(Y, O_{Y} (D′ − C)) ≥ 2.

Without loss of generality we can choose elements x_{1}, x_{2} of that space whose zero set will be a line containing C in ℙ^{3}. Since C is a line it is given by

C: x_{1} = x_{2} = 0.

(2) Since X has a singularity \frac{1}{2}(1, −1) locally at y = 1 it has two local coordinates, say x_{3}, x_{4}, so that the defining equations of X can be written as follows

x_{1}y = −a_{2} and x_{2}y = a_{1},

or equivalently,

y = −\frac{a_{2}}{x_{1}} = \frac{a_{1}}{x_{2}}

where a_{1} and a_{2} are general homogeneous polynomials of degree 3 in k[x_{1}, …, x_{4}]. That means that y is a rational function on Y which has a pole along C. Projecting X away from the point (0, 0, 0, 0, 1) gives

Y : a_{1}x_{1} + a_{2}x_{2} = 0 ⊂ ℙ^{3}.

(B) We begin by constructing a nonsingular K3 surface Y = X(4) of codim 1 in ℙ^{3}, containing the line C: x_{1} = x_{2} = 0 and then obtain a quasismooth K3 surface X = X(3, 3) of codim 2 with A_{1} in ℙ(1, 1, 1, 1, 2) from Y by using birational geometry.

Let x_{1}, …, x_{4} be homogeneous coordinates on ℙ^{3}. Define the linear system L of all homogeneous polynomials of degree 4 on ℙ^{3} containing the line C and take a sufficiently general element f = a_{1}x_{1} + a_{2}x_{2} ∈ L where a_{1} and a_{2} are sufficiently general homogeneous polynomials of degree 3 in k[x_{1}, x_{2}, x_{3}, x_{4}]. Denote by Y the variety defined by the equation f = 0 in ℙ^{3}. The base locus of the linear system L is the line C. By Bertini’s theorem (see [11], Remark III.10.9.2) Y is nonsingular away from C. Therefore it is sufficient to check that Y is nonsingular along C. The singular locus of Y along C is (a_{1}(0, 0, x_{3}, x_{4}) = a_{2}(0, 0, x_{3}, x_{4}) = 0) ⊂ ℙ(x_{3}, x_{4}). But a_{1} and a_{2} are general polynomials, therefore the singular locus is empty. Hence Y is a nonsingular K3 surface in ℙ^{3}, containing the line C. Now set a new generator of degree 2

y = \frac{a_{1}}{x_{2}} = −\frac{a_{2}}{x_{1}}

and define a rational map φ: Y ⊂ ℙ^{3} → X ⊂ ℙ(1, 1, 1, 1, 2), where X is the complete intersection x_{1}y = −a_{2} and x_{2}y = a_{1}, by

(x_{1}, x_{2}, x_{3}, x_{4}) ⟼ (x_{1}, x_{2}, x_{3}, x_{4}, y).

It can easily be observed that φ is a birational map, and in fact an isomorphism Y \ C ˜→ X \ q_{4} where q_{4} = (0, 0, 0, 0, 1). Indeed, the inverse map is projection from q_{4}. Thus it suffices to see that Cone(X) is nonsingular at q_{4}. The rank of the Jacobian matrix of Cone(X) is two at this point. Therefore Cone(X) is nonsingular. The singularities of X arise due to the singularities of ℙ and occur only at the vertex q_{4} which is a singularity of type A_{1} in X.

5.2 Existence of K3 surfaces in codimension 3

We state the main theorem of this section.

Theorem 5.2 The list of 70 candidates is complete and they all exist.

Completeness of this list comes from an exhaustive computer search. For existence we work on model examples from the list. The strategy is as follows:

1. Reduce the codimension of candidates by projecting and then lift back by using birational geometry.

2. From first principles prove nonsingularity.

The seventy candidates will be dealt with as follows (see Table 5.1). The first example in the list is a complete intersection X(2, 2, 2) ⊂ ℙ^{5} and can be done by Bertini’s theorem as explained in Example 5.3. Sixty four candidates can be done by the first type of unprojection and two candidates by the second type of unprojection (see below for definitions). We explain how to construct explicitly quasismooth K3 surfaces of codim 3 from either K3 surfaces of codim 1 or codim 2 by using birational geometry or unprojectionprojection. In the remaining three candidates unprojection-projection is either too complicated or can not be used. We overcome these difficulties by directly proving nonsingularity of a special example using Macaulay.

Example 5.3 Any k general hypersurfaces of ℙ^{n} define an (n−k)-dimensional complete intersection by Bertini’s theorem. For n = 5, to define a surface in ℙ^{5}, we must choose k = 3, and therefore the only c. i. K3 surface in ℙ^{5} is X(2, 2, 2).

Type I unprojection

Let Y be a projectively Gorenstein variety of codimension c embedded in weighted projective space ℙ(a_{0}, …, a_{n}), containing a projectively Gorenstein variety C of codimension c+1. Let I(C) be the ideal of C generated by homogeneous polynomials h_{i} for i = 1, …, l, and I(Y) the ideal of Y generated by f_{i} for i = 1, …, k. Then there exist a rational function v on Y which has a pole along C, that is,

v = \frac{m_{1}}{h_{1}} = ⋯ = \frac{m_{l}}{h_{l}}

for some homogeneous polynomials m_{i} such that X defined by

f_{1} = ⋯ = f_{k} = vh_{1} − m_{1} = ⋯ = vh_{l} − m_{l} = 0

is a projectively Gorenstein variety of codimension c+1 in ℙ(a_{0}, …, a_{n}, a_{n+1}) where wt(v) = a_{n+1}. This construction is called unprojection. (See Introduction or Kustin–Miller [15], Theorem 1.5 and Reid [21]).

Note that a projectively Gorenstein variety means that the projective coordinate ring of the variety is Gorenstein.

Example 5.4 To prove the existence of a quasismooth K3 surface X = X(2, 3, 3, 3, 3) in ℙ(1, 1, 1, 1, 1, 2) with A_{1}, we construct a nonsingular K3 surface Y = X(2, 3) of codim 2 in ℙ^{4} containing the line C: x_{1} = x_{2} = x_{3} = 0.

The first question to ask is how we know that Y and C are the right choices. In the first part of Example 5.1 we show that there are two ways of getting Y and C. We start by assuming that X is given and then we either blow up the singularity and then contract down a suitable configuration or project X away from the singular point q_{5} = (0, 0, 0, 0, 0, 1) to get Y and C. We briefly describe the latter construction. Let x_{1}, …, x_{5}, y be homogeneous coordinates on ℙ(1, 1, 1, 1, 1, 2) with weights 1, 1, 1, 1, 1, 2. Since X has a singularity \frac{1}{2}(1, −1) it has local coordinates, say x_{4}, x_{5}, at q_{5}. In other words, there are three linearly independent equations of X at q_{5}, which are x_{1}y = g_{1}, x_{2}y = g_{2}, x_{3}y = g_{3} for some homogeneous polynomials g_{i} ∈ k[x_{1}, …, x_{5}]. When we write down a 5 × 5 skew-symmetric matrix whose submaximal Pfaffians are the defining equations of X we see that the other two equations of degrees 2, 3 are in k[x_{1}, …, x_{5}] and contain the line C: x_{1} = x_{2} = x_{3} = 0, and we denote Y by the zero locus of these two equations. We project X away from q_{5} to get Y. From now on in all of our examples we write down Y immediately without any explanation. One can work out the details as explained above.

Now we construct X from Y. Let x_{1}, …, x_{5} be homogeneous coordinates on ℙ^{4}. Define linear systems L_{1} and L_{2} of all homogeneous polynomials of degrees 2 and 3 respectively on ℙ^{4}, containing the line C. Let f_{i} = ∑_{j=1}^{3} a_{ij}x_{j} ∈ L_{i} be sufficiently general polynomials for i = 1, 2 where a_{ij}’s are sufficiently general in k[x_{1}, …, x_{5}]. Define

Y : (f_{1} = f_{2} = 0) ⊂ ℙ^{4}.

Both of the linear systems have base locus C so that by Bertini’s theorem, the general members of these linear systems are smooth outside C. To prove that Y is smooth on C, it is sufficient to show that for sufficiently general a_{ij}, the Jacobian matrix of f_{1}, f_{2}, with respect to x_{1}, x_{2}, x_{3}, has rank 2 at every point of the line C, that is,

rk (\begin{matrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{matrix})_{|_{C}} = 2.

For example, take the following matrix on C

(\begin{matrix}0 & x_{5} & x_{4} \\ x_{5}^{2} & x_{4}^{2} & 0 \end{matrix})

which has rank 2 at every point of C = ℙ(x_{4}, x_{5}). Hence Y is nonsingular along C.

We want to find a rational function on Y which has a pole along C. Notice that Y can be given by a system of linear equations

Mx^{t} = 0  (5.2)

where M = (a_{ij}) is a 2 × 3 matrix and x = (x_{1}, x_{2}, x_{3}). Now by Cramer’s rule, we can solve (5.2) to get

yx = ⋀^{2} M

where ⋀^{2} M is the vector of the 2×2 minors of M. Define X by the following five equations

yx = ⋀^{2} M and Mx^{t} = 0

which are also the submaximal Pfaffians of the following 5×5 skew-symmetric matrix

(\begin{matrix}0 & x_{1} & −x_{2} & a_{13} & a_{23} \\ & 0 & x_{3} & a_{12} & a_{22} \\ & & 0 & a_{11} & a_{21} \\ & −sym & & 0 & −y \\ & & & & 0 \end{matrix}) .

Let φ be the rational map φ: Y ⊂ ℙ^{4} → X ⊂ ℙ(1, 1, 1, 1, 1, 2) given by

(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) ⟼ (x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, y).

We claim that Y \C ⟶ X\q_{5}, where q_{5} = (0, 0, 0, 0, 0, 1), is an isomorphism. Restricting φ to Y \ C gives a birational morphism Y \ C → X \ q_{5}. The inverse of φ is projection from q_{5}. This proves the claim. To show that X is quasismooth, it is sufficient to show that the cone is smooth at q_{5}. It can easily be observed that yx = ⋀^{2} M gives three linearly independent equations locally at q_{5}. This implies that X is quasismooth with a singularity A_{1}.

Example 5.5 We show the existence of a quasismooth K3 surface X = X(6, 6, 6, 7, 7)⊂ ℙ(1, 2, 3, 3, 3, 4) with 3A_{2}, A_{3} by constructing a quasismooth K3 surface Y = X(6, 6) ⊂ ℙ(1, 2, 3, 3, 3) with 4A_{2} containing the line C: y = z_{1} = z_{2} = 0.

Let x, y, z_{1}, z_{2}, z_{3} be homogeneous coordinates on ℙ(1, 2, 3, 3, 3) of weights 1, 2, 3, 3, 3 respectively. Let L be the linear system of all homogeneous polynomials of degree 6 containing C with respect to weights 1, 2, 3, 3, 3. Let f_{1}, f_{2} ∈ L be sufficiently general elements, that is,

\begin{aligned}f_{1} & = a_{11}y + a_{12}z_{1} + a_{13}z_{2}, \\ f_{2} & = a_{21}y + a_{22}z_{1} + a_{23}z_{2} \end{aligned}

where a_{ij} are sufficiently general. A general K3 surface Y of codim 2 containing C in ℙ(1, 2, 3, 3, 3) is given by

Mx^{t} = 0  (5.3)

where M is the 2 × 3 matrix (a_{ij}) and x = (y, z_{1}, z_{2}).

By Bertini’s theorem, the singularities of general elements in the linear systems lie on the base locus Cone(C). To show that Y is quasismooth it is sufficient to prove that the cone is smooth along Cone(C) except at the origin. For sufficiently general a_{ij}, we can ensure that the Jacobian matrix of f_{i}, with respect to y, z_{1}, z_{2},

(\begin{matrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{matrix}) along Cone^{∗}(C)

has rank 2. For example, choose

(\begin{matrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{matrix})_{|_{C}} = (\begin{matrix}x^{4} & 0 & z_{3} \\ 0 & z_{3} & x^{3} \end{matrix}) ,

so that the matrix has rank 2 at every point of Cone^{∗}(C). We claim that Y has 4A_{2} singularities. Indeed, the singularities of Y come from the singularities of ℙ and occur only on the vertices, edges and faces of ℙ. It is not difficult to see that the point q_{4} is the only vertex which gives rise to a singularity A_{2}. Now we consider the face q_{2}q_{3}q_{4} of ℙ. The homogeneous polynomials f_{1}, f_{2} on q_{2}q_{3}q_{4} can be written in the homogeneous coordinates z_{1}, z_{2}, z_{3} of ℙ(3, 3, 3). Since ℙ(3, 3, 3) is isomorphic to ℙ^{2} and f_{1}, f_{2} are of degree 2 in ℙ^{2}, by Bézout’s theorem, (f_{1} = f_{2} = 0) in ℙ^{2} consists of exactly four points counted with multiplicity, including the point q_{4}. By the inverse function theorem, x, y are local coordinates and so each point is of type \frac{1}{3}(1, 2). Hence the claim follows.

Now we are ready to construct X with 3A_{2}, A_{3} in ℙ(1, 2, 3, 3, 3, 4) from Y with 4A_{2} in ℙ(1, 2, 3, 3, 3). By Cramer’s rule, we can solve (5.3) to get a new generator t of degree 4 such that ty = m_{23}, tz_{1} = m_{13} and tz_{2} = m_{12} where the m_{ij} are the minors of the 2 × 3 matrix M. This gives three new equations together with f_{1}, f_{2}. Define X by these equations which are just the submaximal Pfaffians of the following 5 × 5 skew-symmetric matrix

(\begin{matrix}0 & t & a_{11} & a_{12} & a_{13} \\ & 0 & a_{21} & a_{22} & a_{23} \\ & & 0 & z_{2} & −z_{1} \\ & −sym & & 0 & y \\ & & & & 0 \end{matrix}) .

The last thing is to show that X has only the four singularities 3A_{2}, A_{3}. Consider the rational map φ: Y ⊂ ℙ(1, 2, 3, 3, 3) → X ⊂ ℙ(1, 2, 3, 3, 3, 4) given by

(x, y, z_{1}, z_{2}, z_{3}) ⟼ (x, y, z_{1}, z_{2}, z_{3}, t).

Restricting φ to Y \C gives an isomorphism Y \C ˜→ X \q_{5}. The inverse map is projection from the point q_{5}. Three singularities 3A_{2} of Y excluding the point (0, 0, 0, 0, 1) correspond to three points p_{i} of X under the isomorphism for i = 1, 2, 3. Since Y \ C ∪ 3A_{2} ≃ X \ p_{1} ∪ p_{2} ∪ p_{3} ∪ q_{5} it is sufficient to check that X is quasismooth at these points p_{i} and q_{5}. We have three Pfaffians, namely ty = m_{23}, tz_{1} = m_{13} and tz_{2} = m_{12}, which are linearly independent locally at q_{5}. This gives rise to a singularity A_{3}. At the p_{i} the Jacobian matrix of the cone has rank 3 because f_{1}, f_{2} and tz_{1} = m_{13} form a linearly independent set locally at these points. This gives three more singularities 3A_{2}. Hence X is a quasismooth K3 surface with 3A_{2}, A_{3} in ℙ(1, 2, 3, 3, 3, 4).

So far we have showed how to get a K3 surface of codim 3 from a K3 surface of codim 2. Now we give an interesting construction of a K3 surface of codim 3 from a K3 surface of codim 1. The main point of this is to get a codim 4 example by a third unprojection.

Example 5.6 We prove the existence of a quasismooth K3 surface X = X(3, 3, 3, 3, 4) in ℙ(1, 1, 1, 1, 2, 2) with 2A_{1}, and containing the line C: x_{1} = x_{2} = x_{3} = y_{2} = 0. For this, we construct Z = X(4) in ℙ^{3}, containing the two disjoint lines C_{1} : x_{1} = x_{2} = 0 and C_{2} : x_{3} = x_{4} = 0 and having an ordinary double point at q_{3} = (0, 0, 0, 1) ∈ C_{1}. As we can see, Z is not quasismooth.

Let x_{1}, x_{2}, x_{3}, x_{4} be homogeneous coordinates on ℙ^{3} and define the linear system L of all homogeneous polynomials of degree 4 on ℙ^{3} containing C_{1} and C_{2}. Let f ∈ L be sufficiently general and Z the variety given by the zero locus of f. The base locus of the linear system is the union of C_{1} and C_{2}, and by Bertini’s theorem the singularities of X lie in that set. The defining equation of Z can be written

f = ax_{1} + bx_{2} = cx_{3} + dx_{4}

where a = a_{1}x_{3} + a_{2}x_{4}, b = b_{1}x_{3} + b_{2}x_{4}, c = a_{1}x_{1} + b_{1}x_{2} and d = a_{2}x_{1} + b_{2}x_{2}. By an appropriate choice of a and b, we can arrange for Z to have q_{3} ∈ C_{1} as an ordinary double point (that is, a double point with nondegenerate tangent cone), and no other singularities. This imposes conditions on a and b: it implies that a and b do not contain x_{4}^{3}, but contain a monomial x_{4}^{2}x_{i} for some i ∈ {1, 2, 3}.

Now set new generators y_{1}, y_{2} of degree 2 as follows:

y_{1} = \frac{b_{1}x_{3} + b_{2}x_{4}}{x_{1}} = −\frac{a_{1}x_{3} + a_{2}x_{4}}{x_{2}} ,
y_{2} = −\frac{a_{2}x_{1} + b_{2}x_{2}}{x_{3}} = \frac{a_{1}x_{1} + b_{1}x_{2}}{x_{4}} .

This gives five relations

y_{1}x_{1} = b_{1}x_{3} + b_{2}x_{4}, y_{1}x_{2} = −(a_{1}x_{3} + a_{2}x_{4}),
y_{2}x_{3} = −(a_{2}x_{1} + b_{2}x_{2}), y_{2}x_{4} = a_{1}x_{1} + b_{1}x_{2},
y_{1}y_{2} = a_{1}b_{2} − b_{1}a_{2}

which are the submaximal Pfaffians of the following matrix:

(\begin{matrix}0 & −x_{1} & x_{2} & −x_{3} & x_{4} \\ & 0 & y_{2} & b_{2} & b_{1} \\ & & 0 & a_{2} & a_{1} \\ & −sym & & 0 & y_{1} \\ & & & & 0 \end{matrix}) .

The variety defined by these five equations is denoted by X in ℙ(1, 1, 1, 1, 2, 2). Since a_{2} and b_{2} do not contain x_{4}^{2}, it contains the line C. Define a rational map φ: Z ⊂ ℙ^{3} → X ⊂ ℙ(1, 1, 1, 1, 2, 2) via

(x_{1}, x_{2}, x_{3}, x_{4}) ⟼ (x_{1}, x_{2}, x_{3}, x_{4}, y_{1}, y_{2}).

This is a birational map and an isomorphism Z \ C_{1} ∪ C_{2} ˜→ X \ C ∪ q_{4} ∪ q_{5} where q_{4} = (0, 0, 0, 0, 1, 0) and q_{5} = (0, 0, 0, 0, 0, 1). Indeed, the inverse map is projection from the line generated by q_{4} and q_{5}. It is enough to check that X is quasismooth along the line C and at the points q_{4} and q_{5}. At q_{4} there are locally three linearly independent equations x_{1} = b_{1}x_{3} + b_{2}x_{4}, x_{2} = −(a_{1}x_{3} + a_{2}x_{4}) and y_{2} = a_{1}b_{2} − b_{1}a_{2}. This gives a singularity A_{1}. Similarly, the equations x_{3} = −(a_{2}x_{1} + b_{2}x_{2}), x_{4} = a_{1}x_{1} + b_{1}x_{2} and y_{1} = a_{1}b_{2} − b_{1}a_{2} are locally linearly independent at q_{5}. So q_{5} is a singular point of type A_{1}. We claim that X is quasismooth along C. To prove the claim, we write the defining equations explicitly

y_{1}x_{1} = x_{4}^{2}(α_{1}x_{1} + β_{1}x_{2} + γ_{1}x_{3}) + x_{4}g_{1}(x_{1}, x_{2}, x_{3}) + g_{2}(x_{1}, x_{2}, x_{3}),
y_{1}x_{2} = −(x_{4}^{2}(α_{2}x_{1} + β_{2}x_{2} + γ_{2}x_{3}) + x_{4}g_{3}(x_{1}, x_{2}, x_{3}) + g_{4}(x_{1}, x_{2}, x_{3})),
y_{2}x_{3} = −(a_{2}x_{1} + b_{2}x_{2}),
y_{2}x_{4} = a_{1}x_{1} + b_{1}x_{2},
y_{1}y_{2} = a_{2}b_{1} − b_{2}b_{1}

where the g_{i} are homogeneous polynomials with the appropriate degrees in x_{1}, x_{2}, x_{3}, and α_{i}, β_{i}, γ_{i} ∈ k for every i. Without loss of generality we can assume that a_{1} and b_{1} do not contain any terms involving x_{4} as a multiple since we can include these terms in a_{2} and b_{2}. If x_{4} = 0 on C then we have only one point q_{4} at which X is quasismooth. So assume x_{4} = 1. The Jacobian matrix J of X along C with respect to the homogeneous coordinates is

(\begin{matrix}α_{1} − y_{1} & β_{1} & γ_{1} & 0 & 0 & 0 \\ α_{2} & β_{2} + y_{1} & γ_{2} & 0 & 0 & 0 \\ a_{2} & b_{2} & y_{2} & 0 & 0 & 0 \\ a_{1} & b_{1} & 0 & 0 & 0 & x_{4} \\ 0 & 0 & 0 & 0 & 0 & y_{1} \end{matrix}) along C.

Now

rk J = 3 ⟺ rk (\begin{matrix}α_{1} − y_{1} & β_{1} & γ_{1} \\ α_{2} & β_{2} + y_{1} & γ_{2} \end{matrix}) = 2

which is equivalent to the 2 × 2 minors of the matrix, denoted by M, on the right hand side having an empty solution. Locally at x_{4} = 1

f = (α_{2}x_{1} + β_{2}x_{2} + γ_{2}x_{3})x_{1} + (α_{1}x_{1} + β_{1}x_{2} + γ_{1}x_{3})x_{2} + ⋯

where ⋯ refers to higher degree terms. Denote by f^{(2)} the homogeneous polynomial of degree 2 in f, that is, the tangent cone of f. This can be written in the form

(x_{1}, x_{2}, x_{3})N(x_{1}, x_{2}, x_{3})^{t}

where

N = (\begin{matrix}α_{2} & \frac{1}{2}(α_{1} + β_{2}) & \frac{1}{2}γ_{2} \\ \frac{1}{2}(α_{1} + β_{2}) & β_{1} & \frac{1}{2}γ_{1} \\ \frac{1}{2}γ_{2} & \frac{1}{2}γ_{1} & 0 \end{matrix}) .

Now det N = \frac{1}{4}(−γ_{1}(α_{2}γ_{1} − β_{2}γ_{2}) + γ_{2}(γ_{1}α_{1} − γ_{2}β_{1})) ≠ 0 since the tangent cone is nondegenerate. This implies that γ_{1} and γ_{2} cannot be zero at the same time. Assume that γ_{1} = 0. Then β_{1} is nonzero since det N ≠ 0, which implies that rk M = 2. Similarly, if γ_{2} = 0 then α_{2} is nonzero and therefore rk M = 2. Now assume that both are nonzero. The 2 × 2 minors of M are

(α_{1} − y_{1})(β_{2} + y_{1}) − β_{1}α_{2} = 0,
(α_{1} − y_{1})γ_{2} − α_{2}γ_{1} = 0,
β_{1}γ_{2} − γ_{1}(β_{2} + y_{1}) = 0

and we note that the first minor is generated by other two minors, namely

(α_{1} − y_{1})(β_{2} + y_{1}) − β_{1}α_{2} =

\frac{(y_{1} + β_{2})}{γ_{2}}((α_{1} − y_{1})γ_{2} − α_{2}γ_{1}) − \frac{α_{2}}{γ_{2}} (β_{1}γ_{2} − γ_{1}(β_{2} + y_{1})).

Now assume that the last two minors have a common solution. Then since γ_{1} and γ_{2} are both nonzero we can multiply the first of these by γ_{1}, the second by −γ_{2} and add together to get 4 det N = 0 which contradicts the assumption. Hence X is nonsingular along C. Therefore it has only two singularities 2A_{1}.

Type II unprojection

The construction we give for the Type II unprojection is based on notes of Reid. We start by constructing a hypersurface Y ⊂ 𝔸^{4} containing a nonnormal curve to obtain a codim 3 Gorenstein X ⊂ 𝔸^{6}.

Let Γ ⊂ 𝔸^{4} be the nonnormal variety parametrised by

x = r^{2}, y = r^{3}, z = s and t = rs.

Its four defining equations can be given by the nongeneric determinantal form

rk (\begin{matrix}x & z & y & t \\ y & t & x^{2} & xz \end{matrix}) ≤ 1.  (5.4)

A general hypersurface Y containing Γ is defined by the zero locus of

f = A(xt − yz) + B(y^{2} − x^{3}) + C(x^{2}z − yt) + D(t^{2} − xz^{2}),

where A, B, C, D are general polynomials in k[x, y, z, t]. By analogy with the Type I unprojection we are looking for rational sections of O_{Y} (n) with a pole along Γ. Consider the following matrix N associated to f:

N = (\begin{matrix}x & y & xC − zD \\ z & t & xB \\ y & x^{2} & xA + tD \\ t & xz & zA − yB + tC \end{matrix}) .

We are not concerned here with where this matrix originally came from. Notice that the 3 × 3 minors of N equal f times x, z, y, t respectively, and that

(t, −y, z, −x)N = (0, 0, 0, 0) and (xz, −x^{2}, t, −y)N = (0, 0, f).

Since the 3×3 minors of N are divisible by f and rk N = 2 on Y the equations

N (\begin{matrix}u \\ v \\ 1 \end{matrix}) = 0 on Y  (5.5)

have a unique solution u, v ∈ K(Y). Clearly, u, v are rational functions which become regular on multiplying by any 2 × 2 minors of the matrix in (5.4).

This construction leads to a codim 3 Gorenstein subvariety X whose defining equations are the submaximal Pfaffians of a 5 × 5 skew-symmetric matrix M where

M = (\begin{matrix}0 & x & z & y & t \\ & 0 & v & D & −u \\ & & 0 & u + C & B \\ & −sym & & 0 & A + xv \\ & & & & 0 \end{matrix})

in k[x, y, z, t, u, v]. In other words, we add u, v to the ring R(Y) of Y in order to get the Gorenstein ring R(Y)[u, v] of codimension 3, the ring of X.

To be more precise, the four submaximal Pfaffians Pf_{i} for i = 2, …, 4 are the four equations in (5.5), and Pf_{1} can be written as a combination of the others since M(Pf_{1}, …, Pf_{5}) = 0. Now consider the map Y ⊂ 𝔸^{4} → X ⊂ 𝔸^{6} via

(x, y, z, t) ⟼ (x, y, z, t, u, v).

This is a birational map and an isomorphism Y \ Γ → X \ q_{5}. Hence codim X = 3. Since X is given by the maximal Pfaffians of M, it is Gorenstein. Because of the isomorphism it is sufficient to check that X is smooth at q_{5}. Since we have the 3 Pfaffians, namely

vy = …, vt = …, and vA = …,

X is smooth at q_{5} if and only if A contains a nonzero linear term in x, z. Now we apply this construction to K3 surfaces.

Example 5.7 We want to construct X = X(7, 8, 8, 9, 10) ⊂ ℙ(2, 3, 3, 4, 4, 5) having singularities 3A_{1}, 3A_{2}, A_{3} via constructing a codim 1 K3 surface Y = X(12) ⊂ ℙ(2, 3, 3, 4) with singularities 3A_{1}, 4A_{2} containing a projectively nonnormal curve Γ.

Let x, y_{1}, y_{2}, z_{1} be homogeneous coordinates on ℙ(2, 3, 3, 4) with weights 2, 3, 3, 4 respectively. The curve Γ is given in parametric form by

x = r^{2}, y_{1} = r^{3}, y_{2} = s, z_{1} = rs.

The defining equations of Γ can be put into determinantal form

rk (\begin{matrix}x & y_{2} & y_{1} & z_{1} \\ y_{1} & z_{1} & x^{2} & xy_{2} \end{matrix}) ≤ 1.

Notice that this is not a generic determinantal form because two of the minors are combinations of others. Now define the linear system L of all homogeneous polynomials of degree 12 with respect to weights 2, 3, 3, 4 containing Γ:

L = {A(xz_{1}−y_{2}y_{1})+B(x^{3}−y_{1}^{2})+C(x^{2}y_{2}−y_{1}z_{1})+D(xy_{2}^{2}−z_{1}^{2})|A, B, C, D are homogeneous polynomials in degrees 6, 6, 5, 4 resp. in k[x, y_{1}, y_{2}, z_{1}]}.

Let f ∈ L be general element, that is,

f = A(xz_{1} − y_{2}y_{1}) + B(x^{3} − y_{1}^{2}) + C(x^{2}y_{2} − y_{1}z_{1}) + D(xy_{2}^{2} − z_{1}^{2})

where A, B, C, D are general homogeneous polynomials. Define a general hypersurface Y ⊂ ℙ(2, 3, 3, 4) by the zero locus of f. By Bertini’s theorem Cone(Y) is nonsingular away from Cone(Γ). Also, Cone(Y) is smooth along Cone(Γ) except at the origin. Under the ℂ^{∗}-action Y has singularities 3A_{1}, 4A_{2}.

Now we write down the 4 × 3 matrix N associated to f:

N = (\begin{matrix}x & y_{1} & xC + y_{2}D \\ y_{2} & z_{1} & −xB \\ y_{1} & x^{2} & xA − z_{1}D \\ z_{1} & xy_{2} & y_{2}A + y_{1}B + z_{1}C \end{matrix})

whose 3 × 3 minors are f times x, y_{1}, y_{2}, z_{1} respectively. The equations

N (\begin{matrix}t \\ z_{2} \\ 1 \end{matrix}) = 0 on Y

have a unique solution t, z_{2} ∈ K(Y) of degrees 5, 4. This gives four equations which are four submaximal Pfaffians of the following matrix:

M = (\begin{matrix}0 & x & y_{2} & y_{1} & z_{1} \\ & 0 & z_{2} & −D & −t \\ & & 0 & t + C & −B \\ & −sym & & 0 & A + xz_{2} \\ & & & & 0 \end{matrix}) .

The fifth equation can be seen from the matrix M or alternatively can be written as a combination of the others since these are not linearly independent. Define X by these five Pfaffians associated to M. The original hypersurface Y can be obtained by eliminating t and z_{2} from the defining equations of X.

To see that X is a quasismooth K3 surface with singularities 3A_{1}, 3A_{2}, A_{3} in ℙ(2, 3, 3, 4, 4, 5) consider a rational map φ: Y → X given by

(x, y_{1}, y_{2}, z_{1}) ⟼ (x, y_{1}, y_{2}, z_{1}, z_{2}, t).

It is a morphism away from Γ and moreover an isomorphism Y \ Γ ˜→ X \ q_{4}. For, we begin with a map

π_{L}: ℙ(2, 3, 3, 4, 4, 5) \ L ⟼ ℙ(2, 3, 3, 4)

which is projection from the line L: x = y_{1} = y_{2} = z_{1} = 0. Restricting π_{L} to X gives a morphism from X \ q_{4} to Y \ Γ which is the inverse of φ. Since Y \ Γ ≃ X \ q_{4} it suffices to check that Cone(X) is smooth at the point q_{4}. Locally at q_{4} Pfaffians Pf_{1}, Pf_{4}, Pf_{5} form a linearly independent set. By the inverse function theorem y_{2}, t are local coordinates. This gives a singularity A_{3}. Similarly, it can be proven that X has the further singularities 3A_{1}, 3A_{2}.

The next example can be done in a similar way. Here we just state the result and give the machinery.

Example 5.8 X(10, 11, 12, 13, 14) ⊂ ℙ(3, 4, 5, 5, 6, 7) with 2A_{2}, A_{3}, A_{4}, A_{4} is a quasismooth K3 surface of codim 3. This follows from first constructing a general hypersurface X(18) ⊂ ℙ(3, 4, 5, 6) with 3A_{2}, A_{3}, A_{1}, A_{4}, containing the nonnormal curve Γ given by nongeneric determinantal form

rk (\begin{matrix}z & y & u & t \\ u & t & z^{2} & yz \end{matrix}) ≤ 1

where y, z, t, u are homogeneous coordinates on ℙ(3, 4, 5, 6) with weights 3, 4, 5, 6 respectively.

Nonsingularity of the remaining three candidates via Macaulay

The remaining three candidates (X, D) in the list are numbers 61, 65 and 70. We will consider just the first of these; the other two can be done in a similar way.

Recall that X is defined by the 5 Pfaffians Pf_{i} in the weighted projective space ℙ(x_{0}, …, x_{5}) with weights wt(x_{i}) = a_{i}. Let J(X) be the Jacobian matrix of X given by a 5 × 6 matrix

J(X) = (\frac{∂ Pf_{i}}{∂x_{i}}).

Denote by Sing X the singular locus of X, which is defined by the 3×3 minors of J(X) together with the ideal I(X) of X. To check the nonsingularity of X it suffices to prove that Sing X = ∅. We will show this by computing the ideal I(Sing X) via Macaulay.

Example 5.9 We consider candidate 61. Let a, b, c, d, e, f be homogeneous coordinates on ℙ(5, 5, 6, 7, 8, 9) with weights 5, 5, 6, 7, 8, 9. The defining equations of X can be represented by a skew symmetric matrix M which, after a suitable change of coordinates, is of the form

M = (\begin{matrix}0 & a & c & d & e \\ −a & 0 & d & e & f \\ −c & −d & 0 & f & a^{2} + ab + b^{2} \\ −d & −e & −f & 0 & bc \\ −e & −f & −a^{2} − ab − b^{2} & −bc & 0 \end{matrix}) .

Here we assume that all of the coefficients of the entries of M are 1. We think of 1 as a generic number. We start a Macaulay session by defining a ring r.

% ring r
characteristic : 0
number of variables : 6
6 variables : a b c d e f
variables weights : 5 5 6 7 8 9
monomial order : reverse lexicographic order
% mat M
number of rows: 5
number of columns: 5
input of entries:
(1,1)=0
(2,1)=-a
     .
     .

The notations I(X), J(X), I(sing X) are not suitable for Macaulay therefore we denote these by IX, JX, ISingX.

% pfaff M IX
% jacob IX JX
% wedge JX 3 N

where N is a 10 × 20 matrix whose entries give the ideal of Sing X.

The command “std” calculates the standard basis of I(Sing X). The command “putstd” allows to view standard basis as a matrix.

% flatten N ISingX
% std ISingX ISingX_std
% putstd ISingX_std ISingX_pstd

We can now observe the standard basis by simply typing:

% type ISingX_pstd

The standard basis contains

f^{3}, b^{6}, a^{5} + 55/6e^{2}f − 7/2df^{2},
d^{3} − 3/2cde − ae^{2} + 1/2c^{2}f − 1/2adf,
e^{3} + 4a^{3}f + 7/2a^{2}bf + 4ab^{2}f + 10def − 13/2cf^{2},
c^{4} + 29/12a^{3}f + 55/12a^{2}bf + 35/12ab^{2}f + 41/6def − 31/4cf^{2}, ….

The zero locus of these equations is empty therefore Sing X is empty, as required. In Example 4.6, we show that (X, D) has the given singularities. This completes the existence of this surface with the given features.

5.2.1 List of K3 surfaces in codimension 3

Table 5.1: Weighted K3 surfaces in codimension 3

Nb.  Weighted K3 surface  Resolution Data
1  X(2, 2, 2) ⊂ ℙ(1, 1, 1, 1, 1, 1)  8
2  X(2, 3, 3, 3, 3) ⊂ ℙ(1, 1, 1, 1, 1, 2)  6 + \frac{1}{2}
3  X(3, 3, 4, 4, 4) ⊂ ℙ(1, 1, 1, 1, 2, 3)  4 + \frac{2}{3}
4  X(3, 3, 3, 3, 4) ⊂ ℙ(1, 1, 1, 1, 2, 2)  4 + 2 × \frac{1}{2}
5  X(3, 3, 4, 4, 4) ⊂ ℙ(1, 1, 1, 2, 2, 2)  2 + 3 × \frac{1}{2}
6  X(4, 4, 5, 5, 6) ⊂ ℙ(1, 1, 1, 2, 3, 4)  2 + \frac{3}{4}
7  X(3, 4, 4, 4, 5) ⊂ ℙ(1, 1, 1, 2, 2, 3)  2 + \frac{1}{2} + \frac{2}{3}
8  X(4, 4, 4, 4, 4) ⊂ ℙ(1, 1, 2, 2, 2, 2)  0 + 5 × \frac{1}{2}

CHAPTER 5. UNPROJECTION-PROJECTION 69

Table 5.1: Weighted K3 surfaces in codimension 3

Nb.  Weighted K3 surface  Resolution Data
9  X(6, 7, 8, 9, 10) ⊂ ℙ(1, 1, 3, 4, 5, 6)  0 + \frac{5}{6}
10  X(4, 4, 4, 5, 5) ⊂ ℙ(1, 1, 2, 2, 2, 3)  0 + 3 × \frac{1}{2} + \frac{2}{3}
11  X(4, 4, 5, 5, 6) ⊂ ℙ(1, 1, 2, 2, 3, 3)  0 + \frac{1}{2} + 2 × \frac{2}{3}
12  X(4, 5, 6, 6, 7) ⊂ ℙ(1, 1, 2, 2, 3, 5)  0 + \frac{1}{2} + \frac{2·3}{5}
13  X(4, 5, 5, 6, 6) ⊂ ℙ(1, 1, 2, 2, 3, 4)  0 + 2 × \frac{1}{2} + \frac{3}{4}
14  X(5, 6, 6, 7, 8) ⊂ ℙ(1, 1, 2, 3, 4, 5)  0 + \frac{1}{2} + \frac{4}{5}
15  X(4, 5, 6, 6, 7) ⊂ ℙ(1, 1, 2, 3, 3, 4)  0 + \frac{2}{3} + \frac{3}{4}
16  X(4, 5, 5, 6, 6) ⊂ ℙ(1, 2, 2, 2, 3, 3) −2 + 5 × \frac{1}{2} + \frac{2}{3}
17  X(6, 6, 7, 7, 8) ⊂ ℙ(1, 2, 2, 3, 4, 5) −2 + 4 × \frac{1}{2} + \frac{4}{5}
18  X(5, 5, 6, 6, 6) ⊂ ℙ(1, 2, 2, 3, 3, 3) −2 + 2 × \frac{1}{2} + 3 × \frac{2}{3}
19  X(6, 7, 8, 9, 10) ⊂ ℙ(1, 2, 2, 3, 5, 7) −2 + 2 × \frac{1}{2} + \frac{3·4}{7}
20  X(8, 9, 10, 11, 12) ⊂ ℙ(1, 2, 4, 5, 6, 7) −2 + 3 × \frac{1}{2} + \frac{6}{7}
21  X(6, 6, 6, 7, 7) ⊂ ℙ(1, 2, 3, 3, 3, 4) −2 + 3 × \frac{2}{3} + \frac{3}{4}
22  X(6, 8, 9, 10, 11) ⊂ ℙ(1, 2, 3, 3, 5, 8) −2 + \frac{2}{3} + \frac{3·5}{8}
23  X(6, 8, 9, 10, 11) ⊂ ℙ(1, 2, 3, 4, 5, 7) −2 + \frac{3}{4} + \frac{3·4}{7}
24 X(10, 11, 12, 13, 14) ⊂ ℙ(1, 3, 5, 6, 7, 8) −2 + 2 × \frac{2}{3} + \frac{7}{8}
25  X(8, 9, 9, 10, 10) ⊂ ℙ(1, 3, 4, 4, 5, 6) −2 + 2 × \frac{3}{4} + \frac{5}{6}
26  X(8, 8, 9, 9, 10) ⊂ ℙ(1, 3, 4, 4, 5, 5) −2 + \frac{3}{4} + 2 × \frac{4}{5}
27 X(14, 15, 16, 17, 18) ⊂ ℙ(1, 5, 7, 8, 9, 10) −2 + \frac{2·3}{5} + \frac{9}{10}
28  X(8, 9, 10, 12, 13) ⊂ ℙ(1, 3, 4, 4, 5, 9) −2 + \frac{3}{4} + \frac{2·7}{9}
29  X(8, 9, 10, 12, 13) ⊂ ℙ(1, 3, 4, 5, 6, 7) −2 + \frac{5}{6} + \frac{2·5}{7}
30 X(16, 17, 18, 19, 20) ⊂ ℙ(1, 6, 8, 9, 10, 11) −2 + \frac{1}{2} + \frac{2}{3} + \frac{10}{11}
31 X(12, 13, 14, 15, 16) ⊂ ℙ(1, 4, 6, 7, 8, 9) −2 + \frac{1}{2} + \frac{3}{4} + \frac{8}{9}

continued on next page

CHAPTER 5. UNPROJECTION-PROJECTION 70

Table 5.1: Weighted K3 surfaces in codimension 3

Nb. Weighted K3 surface Resolution Data
32 X(11, 12, 12, 13, 14) ⊂ ℙ(1, 4, 5, 6, 7, 8) −2 + \frac{1}{2} + \frac{4}{5} + \frac{7}{8}
33 X(10, 11, 12, 12, 13) ⊂ ℙ(1, 4, 5, 6, 6, 7) −2 + \frac{1}{2} + \frac{5}{6} + \frac{6}{7}
34 X(9, 10, 10, 11, 12) ⊂ ℙ(1, 3, 4, 5, 6, 7) −2 + \frac{2}{3} + \frac{3}{4} + \frac{6}{7}
35  X(8, 9, 10, 10, 11) ⊂ ℙ(1, 3, 4, 5, 5, 6) −2 + \frac{2}{3} + \frac{4}{5} + \frac{5}{6}
36  X(6, 6, 7, 7, 8) ⊂ ℙ(1, 2, 3, 3, 4, 4) −2 + \frac{1}{2} + \frac{2}{3} + 2 × \frac{3}{4}
37  X(6, 7, 7, 8, 8) ⊂ ℙ(1, 2, 3, 3, 4, 5) −2 + \frac{1}{2} + 2 × \frac{2}{3} + \frac{4}{5}
38  X(7, 8, 8, 9, 10) ⊂ ℙ(1, 2, 3, 4, 5, 6) −2 + 2 × \frac{1}{2} + \frac{2}{3} + \frac{5}{6}
39  X(6, 7, 8, 8, 9) ⊂ ℙ(1, 2, 3, 4, 4, 5) −2 + 2 × \frac{1}{2} + \frac{3}{4} + \frac{4}{5}
40  X(6, 7, 8, 9, 10) ⊂ ℙ(1, 2, 3, 4, 5, 5) −2 + \frac{1}{2} + \frac{4}{5} + \frac{2·3}{5}
41  X(6, 6, 7, 8, 9) ⊂ ℙ(1, 2, 3, 3, 4, 5) −2 + \frac{2}{3} + \frac{3}{4} + \frac{2·3}{5}
42  X(6, 7, 8, 9, 10) ⊂ ℙ(1, 2, 3, 3, 4, 7) −2 + \frac{1}{2} + \frac{2}{3} + \frac{2·5}{7}
43  X(5, 6, 6, 6, 7) ⊂ ℙ(1, 2, 2, 3, 3, 4) −2 + 3 × \frac{1}{2} + \frac{2}{3} + \frac{3}{4}
44  X(5, 6, 6, 7, 8) ⊂ ℙ(1, 2, 2, 3, 3, 5) −2 + 2 × \frac{1}{2} + \frac{2}{3} + \frac{2·3}{5}
45 X(10, 12, 13, 14, 15) ⊂ ℙ(3, 3, 4, 5, 7, 10) −4 + 3 × \frac{2}{3} + \frac{3·7}{10}
46 X(12, 14, 15, 16, 19) ⊂ ℙ(3, 4, 5, 7, 8, 11) −4 + \frac{3·5}{8} + \frac{3·8}{11}
47 X(10, 12, 14, 15, 17) ⊂ ℙ(2, 3, 5, 7, 8, 9) −4 + \frac{3·5}{8} + \frac{4·5}{9}
48 X(10, 12, 14, 15, 17) ⊂ ℙ(2, 3, 5, 5, 7, 12) −4 + \frac{2·3}{5} + \frac{5·7}{12}
49 X(9, 10, 11, 12, 14) ⊂ ℙ(2, 2, 3, 5, 7, 9) −4 + 4 × \frac{1}{2} + \frac{4·5}{9}
50  X(6, 7, 8, 8, 9) ⊂ ℙ(2, 2, 3, 3, 4, 5) −4 + 5 × \frac{1}{2} + \frac{2}{3} + \frac{2·3}{5}
51 X(14, 15, 16, 17, 18) ⊂ ℙ(2, 5, 6, 7, 9, 11) −4 + \frac{1}{2} + \frac{5}{6} + \frac{5·6}{11}
52 X(9, 10, 11, 12, 14) ⊂ ℙ(2, 3, 4, 5, 7, 7) −4 + 2 × \frac{1}{2} + \frac{3·4}{7} + \frac{2·5}{7}
53 X(12, 13, 14, 15, 16) ⊂ ℙ(2, 5, 6, 7, 7, 8) −4 + 3 × \frac{1}{2} + \frac{6}{7} + \frac{3·4}{7}
54  X(8, 9, 10, 11, 12) ⊂ ℙ(2, 3, 3, 4, 5, 8) −4 + 2 × \frac{1}{2} + 2 × \frac{2}{3} + \frac{3·5}{8}

continued on next page

Table 5.1: Weighted K3 surfaces in codimension 3

Nb.  Weighted K3 surface  Resolution Data
55  X(8, 9, 10, 10, 11) ⊂ ℙ(2, 3, 3, 4, 5, 7) −4 + \frac{1}{2} + 3 × \frac{2}{3} + \frac{3·4}{7}
56  X(8, 9, 9, 10, 10) ⊂ ℙ(2, 3, 4, 4, 5, 5) −4 + 3 × \frac{1}{2} + 2 × \frac{3}{4} + \frac{2·3}{5}
57  X(8, 8, 9, 9, 10) ⊂ ℙ(2, 3, 3, 4, 5, 5) −4 + \frac{1}{2} + 2 × \frac{2}{3} + 2 × \frac{2·3}{5}
58  X(8, 9, 10, 11, 12) ⊂ ℙ(2, 3, 4, 4, 5, 7) −4 + 4 × \frac{1}{2} + \frac{3}{4} + \frac{2·5}{7}
59  X(7, 8, 8, 9, 10) ⊂ ℙ(2, 3, 3, 4, 4, 5) −4 + 3 × \frac{1}{2} + 3 × \frac{2}{3} + \frac{3}{4}
60 X(14, 15, 16, 17, 18) ⊂ ℙ(3, 5, 6, 7, 8, 11) −4 + \frac{2}{3} + \frac{5}{6} + \frac{4·7}{11}
61 X(14, 15, 16, 17, 18) ⊂ ℙ(5, 5, 6, 7, 8, 9) −4 + \frac{4}{5} + 2 × \frac{2·3}{5} + \frac{5}{6}
62 X(12, 13, 14, 15, 16) ⊂ ℙ(4, 4, 5, 6, 7, 9) −4 + 2 × \frac{1}{2} + 2 × \frac{3}{4} + \frac{2·7}{9}
63 X(12, 13, 14, 15, 16) ⊂ ℙ(3, 4, 5, 6, 7, 10) −4 + \frac{1}{2} + \frac{2}{3} + \frac{4}{5} + \frac{3·7}{10}
64 X(10, 11, 12, 13, 14) ⊂ ℙ(3, 4, 5, 5, 6, 7) −4 + 2 × \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \frac{2·3}{5}
65 X(12, 13, 14, 15, 16) ⊂ ℙ(4, 5, 5, 6, 7, 8) −4 + \frac{1}{2} + \frac{3}{4} + 2 × \frac{4}{5} + \frac{2·3}{5}
66 X(10, 11, 12, 12, 13) ⊂ ℙ(3, 4, 4, 5, 6, 7) −4 + \frac{1}{2} + \frac{2}{3} + 2 × \frac{3}{4} + \frac{2·5}{7}
67 X(10, 11, 12, 13, 14) ⊂ ℙ(2, 3, 4, 5, 7, 9) −4 + \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4·5}{9}
68 X(9, 10, 10, 11, 12) ⊂ ℙ(2, 3, 4, 5, 5, 7) −4 + \frac{1}{2} + \frac{3}{4} + \frac{2·3}{5} + \frac{3·4}{7}
69  X(8, 9, 10, 11, 12) ⊂ ℙ(2, 3, 4, 5, 5, 6) −4 + 3 × \frac{1}{2} + \frac{2}{3} + \frac{4}{5} + \frac{2·3}{5}
70 X(16, 17, 18, 19, 20) ⊂ ℙ(5, 6, 7, 8, 9, 10) −4 + \frac{1}{2} + \frac{2}{3} + \frac{4}{5} + \frac{2·3}{5} + \frac{6}{7}

5.3 Some examples in codimension 4

In this section we give two types of constructions of K3 surfaces (X, D) in codim 4. Namely, the first construction is through unprojection and the second through an (X, 2D)-model. The latter construction is explained in Example 5.14.

We state an important remark, pointed out by Reid.

Remark 5.10 Let X be a projectively Gorenstein subscheme of codim 3, containing a projectively Gorenstein subscheme C of codim 4, that is the canonical sheaves of X and C are ω_{X} = O_{X}(k_{X}) and ω_{C} = O_{C}(k_{C}) for some k_{X}, k_{C} ∈ ℤ respectively. Then there exists a homomorphism s: I_{C} → O(−l) which is a basis of Hom(I_{C}, O_{X}(−l)) at every point of C if l = k_{C} − k_{X} < 0.

Note that this remark can also be stated for any codimension, in particular, to get codim ≥ 5 examples. Moreover, it leads to the existence of codim ≥ 4 rings but it does not explicitly give the defining equations of the surfaces we construct. In the case of codim 4 we work on particular examples under some mild conditions, which we can give explicitly by defining equations, and furthermore by using birational geometry we show that the surfaces we construct are quasismooth.

The next example leads to the general construction of codim 4 K3 surfaces with explicit equations from successive constructions in codimensions 2 and 3. 112 examples in Table 5.2 can also be obtained in this way.

Example 5.11 We want to construct a codim 4 K3 surface X having A_{6} in ℙ(1, 1, 3, 4, 5, 6, 7) by constructing successively codim 2 and codim 3 K3 surfaces.

Let x_{1}, x_{2}, z, t, u be homogeneous coordinates on ℙ(1, 1, 3, 4, 5) with weights 1, 1, 3, 4, 5 respectively. Now define two linear systems L_{1}, L_{2} of all homogeneous polynomials of degrees 6, 8 respectively in k[x_{1}, x_{2}, z, t, u] with respect to weights 1, 1, 3, 4, 5, containing the line

C_{1} : x_{1} = z = t = 0.

Let f_{i} = x_{1}a_{i1} + za_{i2} + ta_{i3} ∈ L_{i} be sufficiently general for i = 1, 2 where the a_{ij} are sufficiently general homogeneous polynomial in k[x_{1}, x_{2}, z, t, u]. Define a general surface

Z = X(6, 8): (f_{1} = f_{2} = 0) ⊂ ℙ(1, 1, 3, 4, 5).  (5.6)

By Bertini’s theorem the singularities of Cone(Z) lie on the base locus Cone(C_{1}). We want Z to have an ordinary double point q_{1} = (0, 1, 0, 0, 0) and A_{4} along C_{1} where A_{4} corresponds to the point q_{4} = (0, 0, 0, 0, 1). Obviously, this puts some restrictions on a_{ij}’s. Clearly, locally at q_{4} we have

f_{1} = x_{1} + ⋯ and f_{2} = z + ⋯ ⊂ 𝔸^{4} /⟨ε⟩

where ε is a primitive 5th root of unity and ⋯ refers to other terms. By the inverse function theorem x_{2}, t are local coordinates. This implies that q_{4} is a singularity of type A_{4}. The surface Z having a singularity q_{1} is equivalent to rk J(Z)_{|_{q_{1}}} ≤ 1 where the Jacobian matrix J(Z) of Z along C_{1} is

\frac{∂(f_{1}, f_{2})}{∂(x_{1}, z, t)}_{|_{C_{1}}} = (\begin{matrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{matrix})_{|_{C_{1}}} .

Since a_{ij}’s are general rk J(Z)_{|_{q_{1}}} = 1. By replacing f_{2} by f_{2} with a suitable multiple of x_{2}^{2}f_{1} subtracted, we can assume that the a_{2j} do not contain any pure powers of x_{2}. From here it can be observed that Cone(Z) is nonsingular on Cone(C_{1}) except at the origin and q_{1}.

Now by Cramer’s rule we can solve (5.6) to get vx = ⋀^{2} M where M = (a_{ij}) is a 2 × 3 matrix and x = (x_{1}, z, t). Define Y = X(6, 7, 8, 9, 10) ⊂ ℙ(1, 1, 3, 4, 5, 6) by the five equations

vx = ⋀^{2} M and Mx^{t} = 0

which are also the submaximal Pfaffians of the following 5×5 skew-symmetric matrix

N = (\begin{matrix}0 & −t & z & a_{21} & −a_{11} \\ & 0 & −x_{1} & a_{22} & −a_{12} \\ & & 0 & a_{23} & −a_{13} \\ & −sym & & 0 & −v \\ & & & & 0 \end{matrix})

where the entries of N are taken in the ring k[x_{1}, x_{2}, z, t, u, v]. We want to show that Y is a quasismooth K3 surface with A_{5}, containing the line

C_{2} : x_{1} = z = t = u = 0

in ℙ(1, 1, 3, 4, 5, 6). Since x_{2}^{α} ∉ a_{2j} for any nonzero α, Y contains the line C_{2}. Define a rational map φ: Z → Y by

(x_{1}, x_{2}, z, t, u) ⟼ (x_{1}, x_{2}, z, t, u, v).

The map φ restricted to Z\C_{1} gives an isomorphism Z\C_{1} ˜→ Y \C_{2}. Indeed, consider the map from Y \ q_{5} to Z, which is projection from the point q_{5}, and so the inverse map is the restriction of this map to Y \ C_{2}. To see that Cone(Y) is smooth it is enough to consider Cone(Y) on Cone(C_{2}) excluding the origin since Cone(Z) is nonsingular outside Cone(C_{1}). Now observe that (f_{1} = 0) is nonsingular at q_{1} and (f_{2} = 0) is singular at q_{1}. Consider (f_{2} = 0) on (f_{1} = 0) and calculate the tangent cone of (f_{2} = 0) on (f_{1} = 0) at q_{1}. It suffices to show that Cone(Y) being smooth corresponds to the tangent cone of (f_{2} = 0) on (f_{1} = 0) at q_{1} being nondegenerate. One can prove this by using the same argument as that used in Example 5.6. Locally at q_{5} there are three linearly independent equations of Y, which give rise to a singularity A_{5}. Remark 5.10 says there exists a generator w of degree 7 which has a pole along C_{2}. This means that

w = \frac{A}{x_{1}} = \frac{B}{z} = \frac{C}{t} = \frac{D}{u}

is a rational function of weight 7 on Y, having a pole along C_{2} for some homogeneous polynomials A, B, C, D ∈ k[x_{1}, x_{2}, z, t, u, v]. This gives four relations, namely:

x_{1}w = A, zw = B, tw = C and uw = D.

Now denote by X the variety given by the nine equations:

vx = ⋀^{2} M, Mx^{t} = 0 and yw = (A, B, C, D)

where x = (x_{1}, z, t) and y = (x_{1}, z, t, u). Notice that the first five equations are the defining equations of Y. Since a_{2j} does not contain a pure power of x_{2} for j = 1, 2, 3 we can write

a_{2j} = x_{1}λ_{1}^{j} + zλ_{2}^{j} + tλ_{3}^{j} + uλ_{4}^{j}

where λ_{i}^{j} ∈ k[x_{1}, x_{2}, z, t, u] for i = 1, 2, 3, 4. To find A, B, C, D explicitly we rewrite the defining equations Pf_{k} of Y in terms of λ_{i}^{j} by substituting for the a_{2j}:

(Pf_{1}, Pf_{2}, Pf_{3}, Pf_{4}, Pf_{5}) = yH

where

H = (\begin{matrix}v + a_{13}λ_{1}^{2} − a_{12}λ_{1}^{3} & a_{13}λ_{1}^{1} − a_{11}λ_{1}^{3} & a_{12}λ_{1}^{1} − a_{11}λ_{1}^{2} & a_{11} & a_{21} \\ a_{13}λ_{2}^{2} − a_{12}λ_{2}^{3} & a_{13}λ_{2}^{1} − a_{11}λ_{2}^{3} − v & a_{12}λ_{2}^{1} − a_{11}λ_{2}^{2} & a_{12} & a_{22} \\ a_{13}λ_{3}^{2} − a_{12}λ_{3}^{3} & a_{13}λ_{3}^{1} − a_{11}λ_{3}^{3} & a_{12}λ_{3}^{1} − a_{11}λ_{3}^{2} + v & a_{13} & a_{23} \\ a_{13}λ_{4}^{2} & a_{13}λ_{4}^{1} & a_{12}λ_{4}^{1} − a_{11}λ_{4}^{2} & 0 & 0 \end{matrix}) .

First we begin by finding A and B. Since zA−x_{1}B ∈ I(Y), some combination β_{1} Pf_{1} +β_{2} Pf_{1} of Pf_{1} and Pf_{2} can be made to contain a term which is a multiple of a_{13}t. Then we can add a multiple of Pf_{4} to β_{1} Pf_{1} +β_{2} Pf_{1} to make it zero along x_{1} = z = 0. This is possible since Pf_{4} is zero along C_{1}. Explicitly:

\begin{matrix}−(a_{13}λ_{4}^{1}) Pf_{1} +( & a_{13}λ_{4}^{2}) Pf_{2} −(−λ_{4}^{1}(a_{13}λ_{3}^{2} − a_{12}λ_{3}^{3}) \\ + λ_{4}^{2}(a_{13}λ_{3}^{1} − a_{11}λ_{3}^{3})) Pf_{4} = x_{1}A + zB \end{matrix}

where

A =a_{13}λ_{4}^{1}(a_{13}λ_{2}^{2} − a_{12}λ_{2}^{3} − a_{12}λ_{3}^{2}) − a_{13}λ_{4}^{2}(a_{13}λ_{2}^{1} − a_{11}λ_{2}^{3} − v − a_{12}λ_{3}^{1})
+ a_{12}λ_{3}^{3}(a_{12}λ_{4}^{1} − a_{11}λ_{4}^{2}),
B = − a_{13}λ_{4}^{1}(v + a_{13}λ_{1}^{2} − a_{12}λ_{1}^{3} − a_{11}λ_{3}^{2}) + a_{13}λ_{4}^{2}(a_{13}λ_{1}^{1} − a_{11}λ_{1}^{3} − a_{11}λ_{3}^{1})
+ a_{11}λ_{3}^{3}(−a_{12}λ_{4}^{1} + a_{11}λ_{4}^{2}).

Similarly,

\begin{matrix}(−a_{11}λ_{4}^{2} + a_{12}λ_{4}^{1}) Pf_{1} − & (a_{13}λ_{4}^{2}) Pf_{3} +(λ_{4}^{1}(−a_{13}λ_{2}^{2} + a_{12}λ_{2}^{3}) \\ + λ_{4}^{2}(−a_{11}λ_{2}^{3} + a_{13}λ_{2}^{1})) Pf_{4} = x_{1}C − tA \end{matrix}

and

(−a_{13}λ_{2}^{1} + a_{11}λ_{2}^{3} + v + a_{12}λ_{3}^{1}) Pf_{1} +(a_{13}λ_{2}^{2} − a_{12}λ_{2}^{3} − a_{12}λ_{3}^{2}) Pf_{2}
+ (a_{12}λ_{3}^{3}) Pf_{3} +(E) Pf_{4} = −x_{1}D + uA

where

C =(−a_{11}λ_{4}^{2} + a_{12}λ_{4}^{1})(v + a_{13}λ_{1}^{2} − a_{12}λ_{1}^{3} + a_{11}λ_{2}^{3})
− a_{13}λ_{4}^{2}(a_{12}λ_{1}^{1} − a_{11}λ_{1}^{2} − a_{11}λ_{2}^{1}) − a_{13}λ_{4}^{1}(λ_{2}^{2}a_{11}),
D = − (−a_{13}λ_{2}^{1} + a_{11}λ_{2}^{3} + v + a_{12}λ_{3}^{1})(v + a_{13}λ_{1}^{2} − a_{12}λ_{1}^{3})
− (a_{13}λ_{2}^{2} − a_{12}λ_{2}^{3} − a_{12}λ_{3}^{2})(a_{13}λ_{1}^{1} − a_{11}λ_{1}^{3})
− (a_{12}λ_{3}^{3})(a_{12}λ_{1}^{1} − a_{11}λ_{1}^{2}) − a_{11}(−λ_{3}^{1}(−a_{12}λ_{2}^{3} + a_{13}λ_{2}^{2})
+ λ_{3}^{2}(−a_{11}λ_{2}^{3} + a_{13}λ_{2}^{1} − v) − λ_{3}^{3}(−a_{11}λ_{2}^{2} + a_{12}λ_{2}^{1})),
E = − λ_{3}^{1}(−a_{12}λ_{2}^{3} + a_{13}λ_{2}^{2}) + λ_{3}^{2}(−a_{13}λ_{2}^{3} + a_{13}λ_{2}^{1} − v) − λ_{3}^{3}(−a_{11}λ_{2}^{2} + a_{12}λ_{2}^{1}).

We want to see that X is a quasismooth K3 surface with A_{6}. Clearly, X has a singularity A_{6} since locally at this point q_{6} there are four linearly independent equations

x_{1} = A, z = B, t = C and u = D.

Consider the rational map φ: Y → X given by

(x_{1}, x_{2}, z, t, u, v) ⟼ (x_{1}, x_{2}, z, t, u, v, w).

This is a birational map and an isomorphism Y \ C_{2} ˜→ X \ q_{6}. In fact, the inverse map is projection from the point q_{6}.

Example 5.12 We want to construct a quasismooth K3 surface Z of codim 4 with singularities A_{1}, A_{2} in ℙ(1, 1, 1, 1, 2, 2, 3) by constructing two subrings of the graded ring R(X, D), corresponding to two K3 surfaces X, Y of codim 3.

Recall that in Example 5.6 a K3 surface X = X(3, 3, 3, 3, 4) with 2A_{1}, containing the line

C: x_{1} = x_{2} = x_{3} = y_{2} = 0,

in ℙ(1, 1, 1, 1, 2, 2) is constructed via X(4) in ℙ^{3} containing two distinct lines

C_{1} : x_{1} = x_{2} = 0, C_{2} : x_{3} = x_{4} = 0

and the ordinary double point (0, 0, 0, 1). Since X and C ≅ ℙ(1, 2) are projectively Gorenstein and their canonical divisors ω_{X} = O_{X} and ω_{C} = O_{C}(−3), by Remark 5.10 there exists a generator z: I_{C} → O(3) in degree 3. This implies that z has a pole along C which gives four relations f_{6}, …, f_{9}. Denote by Z V (f_{1}, …, f_{9}) where the first five relations f_{1}, …, f_{5} are the defining equations of X. In this case, to check that Z is quasismooth we have to know defining equations of Z explicitly. To see this, we consider the same X(4) as in Example 5.6 to construct a different K3 surface Y = X(3, 3, 4, 4, 4) with A_{2} in ℙ(1, 1, 1, 1, 2, 3) via a K3 surface of codim 2. The generator

y_{1} = \frac{a}{x}_{2} = −\frac{b}{x}_{1}

which gives a K3 surface X(3, 3) with A_{1} in ℙ(1, 1, 1, 1, 2), containing the line

C_{3} : x_{1} = x_{2} = x_{3} = 0

since a and b do not contain x_{4}^{3}. So rewrite the defining equations f_{1}, f_{2} of X(3, 3)

y_{1}x_{2} = n_{1}x_{1} + n_{2}x_{2} + n_{3}x_{3},
y_{1}x_{1} = m_{1}x_{1} + m_{2}x_{2} + m_{3}x_{3} respectively

(5.7)

where a = n_{1}x_{1} + n_{2}x_{2} + n_{3}x_{3}, −b = m_{1}x_{1} + m_{2}x_{2} + m_{3}x_{3}, and n_{i}, m_{i} are homogeneous polynomials of degree 2 in k[x_{1}, x_{2}, x_{3}, x_{4}]. Now by Cramer’s rule we can solve (5.7) to get

zx = ⋀^{2} M

where x = (x_{1}, x_{2}, x_{3}) and M is a 2 × 3 matrix such that Mx^{t} = (f_{1}, f_{2}). These three equations zx = ⋀^{2} M together with the equations f_{1}, f_{2} give a K3 surface Y = X(3, 3, 4, 4, 4) with A_{2} in ℙ(1, 1, 1, 1, 2, 3).

So far we have got eight equations explicitly and now consider all these together with choosing n_{1}, n_{2}, m_{1}, m_{2} divisible by x_{4}, and n_{3} = a_{1}, m_{3} = b_{1} such that n_{1}x_{1} + n_{2}x_{2} + n_{3}x_{3} = a_{1}x_{3} + a_{2}x_{4} and m_{1}x_{1} + m_{2}x_{2} + m_{3}x_{3} = b_{1}x_{3} + b_{2}x_{4}. Multiply y_{2}x_{4} = a_{1}x_{1} + b_{1}x_{2} by z and substitute for zx_{1} and zx_{2} as appropriate to deduce that the last equation f_{9} is zy_{2} = ((a_{1}m_{2} − b_{1}n_{2})a_{1} + (a_{1}m_{1} − b_{1}n_{1})b_{1})/x_{4}.

Define a rational map φ: X ⊂ ℙ(1, 1, 1, 1, 2, 2) → Z ⊂ ℙ(1, 1, 1, 1, 2, 2, 3) via

(x_{1}, x_{2}, x_{3}, x_{4}, y_{1}, y_{2}) ⟼ (x_{1}, x_{2}, x_{3}, x_{4}, y_{1}, y_{2}, z).

This is a birational map and an isomorphism X \ q_{4} ∪ q_{5} ∪ C ˜→ Z \ q_{5} ∪ q_{6} where q_{4}, q_{5} are vertices in ℙ(1, 1, 1, 1, 2, 2) and q_{5}, q_{6} are vertices in ℙ(1, 1, 1, 1, 2, 2, 3). Since X \ q_{4} ∪ q_{5} ∪ C is nonsingular it is sufficient to prove that Z is quasismooth at q_{5}, q_{6}. Locally at q_{5} f_{3}, f_{4}, f_{5}, f_{9} are linearly independent which gives a singularity A_{1}. Locally at q_{6} f_{6}, …, f_{9} are linearly independent. This gives rise to a singularity A_{2}.

The following example is a construction of K3 surface of codim 4 from a K3 surface of codim 2. This type of unprojection allows us to use Bertini’s theorem on codimension 2 K3 surfaces to prove quasismoothness of codimension 4 K3 surfaces.

Example 5.13 Let x_{1}, …, x_{5} be homogeneous coordinates on ℙ^{4}. We begin by constructing a nonsingular K3 surface X(2, 3) containing two distinct lines

C_{1} : x_{1} = x_{2} = x_{3} = 0, C_{2} : x_{3} = x_{4} = x_{5} = 0

in ℙ^{4}. From here we construct a K3 surface X of codim 4 with 2A_{1} in ℙ(1, 1, 1, 1, 1, 2, 2).

Define linear systems L_{1}, L_{2} of all homogeneous polynomials of degrees 2, 3 respectively on ℙ^{4}, containing the lines C_{1} and C_{2} and take f_{1} ∈ L and f_{2} ∈ L sufficiently general, that is,

\begin{aligned}f_{1} & = a_{11}x_{1} + a_{12}x_{2} + a_{13}x_{3}, \\ f_{2} & = a_{21}x_{1} + a_{22}x_{2} + a_{23}x_{3} \end{aligned}  (5.8)

or, equivalently,

\begin{aligned}f_{1} & = b_{11}x_{3} + b_{12}x_{4} + b_{13}x_{5}, \\ f_{2} & = b_{21}x_{3} + b_{22}x_{4} + b_{23}x_{5} \end{aligned}  (5.9)

where a_{ij} and b_{ij} are sufficiently general. Define

Y = X(2, 3): (f_{1} = f_{2} = 0) ⊂ ℙ^{4}.

The linear systems L_{i} have the base locus C_{1}∪C_{2} and so by Bertini’s theorem Y is nonsingular away from C_{1} ∪ C_{2}. Choosing a_{ij}, b_{ij} sufficiently general we can arrange Y nonsingular along C_{1} ∪ C_{2}. It is a K3 surface containing two lines C_{1}, C_{2}.

Now by using Cramer’s rule we can solve (5.8) and (5.9) to get

y_{1}x^{1} = ⋀^{2} M and y_{2}x^{2} = ⋀^{2} N

where M = (a_{ij}), N = (b_{ij}) are 2 × 3 matrices and x^{1} = (x_{1}, x_{2}, x_{3}), x^{2} = (x_{3}, x_{4}, x_{5}). Let H be a hyperplane section of Y. Since Y is nonsingular y_{1} ∈ H^{0}(Y, 2H + C_{1}) and y_{2} ∈ H^{0}(Y, 2H + C_{2}) but y_{1} and y_{2} have zero along C_{2} and C_{1} respectively. Therefore y_{1} ∈ H^{0}(Y, 2H + C_{1} − C_{2}) and y_{2} ∈ H^{0}(Y, 2H + C_{2} − C_{1}) which imply that y_{1}y_{2} ∈ H^{0}(Y, 4H). In other words, y_{1}y_{2} = g(x_{1}, …, x_{5}) for some homogeneous polynomial g of degree 4. The variety X can be defined by the following nine equations:

f_{1} = f_{2} = 0, y_{1}x^{1} = ⋀^{2} M, y_{2}x^{2} = ⋀^{2} N and y_{1}y_{2} = g(x_{1}, …, x_{5}).

Define a rational map φ: Y ⊂ ℙ^{4} → X ⊂ ℙ(1, 1, 1, 1, 1, 2, 2) by sending

(x_{1}, …, x_{5}) ⟼ (x_{1}, …, x_{5}, y_{1}, y_{2}).

This is a birational map and an isomorphism

Y \ C_{1} ∪ C_{2} ˜→ X \ q_{5} ∪ q_{6}.

Hence it is enough to check that X is quasismooth at the vertices q_{5}, q_{6}. Locally at q_{5} there are four linearly independent equations x^{1} = ⋀^{2} M and y_{2} = g(x_{1}, …, x_{5}). This gives a singularity A_{1}. Similarly, at q_{6} there are four linearly independent equations x^{2} = ⋀^{2} N and y_{2} = g(x_{1}, …, x_{5}). Hence q_{6} is of type A_{1}.

The following example is a beautiful construction of a K3 surface (X, D) of codim 4 via an (X, 2D)-model.

Example 5.14 We want to construct a codim 4 K3 surface (X, D) with 7A_{1} in ℙ(1, 2, 2, 2, 2, 3, 3) via constructing an (X, 2D)-model with 7A_{1} in ℙ^{4}. Generally speaking, we construct a complete intersection K3 surface X(2, 3) in ℙ^{4} such that X(2, 3) is tangent to the hyperplane y_{0} = 0 along a twisted cubic curve Γ (by analogy of the famous Kummer surface: begin by taking a plane in ℙ^{3} with a conic C and a quartic surface X(4) in ℙ^{3} which touches the plane along C then X(4) in general has six nodes. This leads to a K3 surface X(4, 6) ⊂ ℙ(1, 2, 2, 2, 3) with six nodes, see the Appendix of this chapter), and div y_{0} is Cartier which is 2Γ with Γ twisted cubic Weil divisor. It turns out that X(2, 3) has seven nodes along Γ. This gives that X(2, 3) is an (X, 2D)-model with 7A_{1} in ℙ^{4}. Then the ring R(X, 2D) can be written down quite simple so that we can figure out the details of the ring R(X, D).

Let y_{1}, …, y_{4} be homogeneous coordinates on ℙ^{3}. The twisted cubic curve Γ′ in ℙ^{3} is given by the quadric equations

Q_{1} : y_{1}y_{3} − y_{2}^{2} = 0, Q_{2} : y_{1}y_{4} − y_{2}y_{3} = 0, Q_{3} : y_{2}y_{4} − y_{3}^{2} = 0.

Let F be a cubic hypersurface in ℙ^{3} given by the equation

y_{1}y_{4}^{2} − 2y_{2}y_{3}y_{4} + y_{3}^{3} = 0.

Observe that Q_{1} ∩ F = 2Γ′ by multiplying the defining equation of F by y_{1} and adding −y_{3}^{2} times the defining equation of Q_{1}. Define a general surface X = X(2, 3) in terms of F, Q_{1} as follows

X: (f_{1} = f_{2} = 0) ⊂ ℙ^{4}

where

f_{1} = y_{1}y_{3} − y_{2}^{2} + y_{0}y_{4}, f_{2} = y_{1}y_{4}^{2} − 2y_{2}y_{3}y_{4} + y_{3}^{3} + y_{0}G

for a general homogeneous polynomial G of degree 2 in the variables y_{0}, …, y_{4}. Then the hyperplane section X ∩(y_{0} = 0) is 2Γ where Γ = Γ′ ∩(y_{0} = 0). Here we set 2D = 2Γ. Notice that since 2D is a hyperplane section, it is Cartier on X. We claim that X has seven nodes along Γ. By Bertini’s theorem, the general X is nonsingular away from Γ. To view the singularities we write the Jacobian matrix J(X) of X along Γ as follows

\frac{∂(f_{1}, f_{2})}{∂(y_{0}, …, y_{4})}|_{Γ} = (\begin{matrix}y_{4} & y_{3} & −2y_{2} & y_{1} & 0 \\ G & y_{4}^{2} & −2y_{3}y_{4} & y_{3}^{2} & 0 \end{matrix})_{|_{Γ}} .

Consider the twisted cubic Γ′ ⊂ ℙ^{3} given as the image of the map

t ⟼ (1, t, t^{2}, t^{3}).

Substitute y_{1} = 1, y_{2} = t, y_{3} = t^{2}, y_{4} = t^{3} into J(X) to get

(\begin{matrix}t^{3} & t^{2} & −2t & 1 & 0 \\ G_{|_{Γ}} & t^{6} & −2t^{5} & t^{4} & 0 \end{matrix}) .  (5.10)

By multiplying the first row of the matrix in (5.10) by −t^{4} and adding this to the second row it leads to the solution of the polynomial G(0, 1, t, t^{2}, t^{3}) − t^{7} of degree 7 in the variable t. This in general gives seven singularities. To see that these are actually nodes it is enough to consider only one singularity, and by a change of coordinates we assume that the singular point is p = (0, 1, 0, 0, 0). This implies that y_{1}^{2} ∉ G. We can easily see that (f_{1} = 0) is nonsingular at p. Therefore (f_{1} = 0) on the affine piece y_{1} = 1 of ℙ^{4} is locally isomorphic to ℂ^{3} and the local coordinates are y_{0}, y_{2}, y_{4}. Locally (f_{2} = 0) on (f_{1} = 0) is (y_{4} − y_{2}^{3})^{2} + y_{0}A by substituting y_{3} = y_{2}^{2} − y_{0}y_{4} where A = y_{2}y_{4}^{2} + y_{0}y_{2}^{4}y_{4} + y_{0}^{2}y_{2}^{2}y_{4} + y_{0}^{3}y_{4}^{3} + G. Proving that the singular point p is a node is equivalent to showing that

(t^{7} − G)(p) = 0 and \frac{∂G}{∂y_{2}} (p) ≠ 0.

Since G is general it contains the term y_{1}y_{2}. Therefore p is a node.

We can blow up the singularities of X along D since 2D is Cartier. Then the resolution σ: S → X along D, by Proposition 2.4, looks like

σ^{∗}(D) = B + \frac{1}{2} ∑_{i=1}^{7} C_{i}.

where the C_{i} are −2-curves and B is the divisor meeting the C_{i} transversally. This implies together with (2D)^{2} = 6 that B^{2} = −2. Then h^{0}(S, σ^{∗}(D)) = h^{0}(X, D) = 1 which implies that H^{0}(X, D) is generated by one element, say x, of degree 1 such that x vanishes on D. As we know, by construction H^{0}(X, 2D) is generated by y_{0}, …, y_{4} such that y_{0} vanishes on Γ. So x^{2} = y_{0}. Consider elements of H^{0}(X, 3D) as elements of H^{0}(X, 4D) vanishing on Γ. This implies the existence of two new generators z_{1}, z_{2} in degree 3 such that

\begin{matrix}R_{3} : & xz_{2} − y_{2}y_{4} + y_{3}^{2} = 0, \\ R_{4} : & xz_{1} − y_{1}y_{4} + y_{2}y_{3} = 0, \\ R_{5} : & y_{2}^{2} − y_{1}y_{3} − y_{0}y_{4} = 0. \end{matrix}  (5.11)

So far we have seven generators x, y_{1} …, y_{4}, z_{1}, z_{2} of an (X, D)-model with three relations in degree 4. We want to prove that (X, D) is a codim 4 K3 surface in ℙ(1, 2, 2, 2, 2, 3, 3) with nine equations. First of all, we find out explicitly these nine equations. Then we prove that we do not need any further ring generators and relations.

Now observe that Γ′ has two syzygies, namely

y_{4}(y_{1}y_{3} − y_{2}^{2}) − y_{3}(y_{1}y_{4} − y_{2}y_{3}) + y_{2}(y_{2}y_{4} − y_{3}^{2}) = 0,
y_{3}(y_{1}y_{3} − y_{2}^{2}) − y_{2}(y_{1}y_{4} − y_{2}y_{3}) + y_{1}(y_{2}y_{4} − y_{3}^{2}) = 0.

By inspecting these two syzygies of Γ′ and the three relations R_{3}, R_{4}, R_{5} of (X, D) we get two more relations in degree 5

\begin{matrix}R_{1} : & y_{2}z_{2} − z_{1}y_{3} − xy_{4}^{2} = 0, \\ R_{2} : & y_{1}z_{2} − z_{1}y_{2} − xy_{4}y_{3} = 0. \end{matrix}  (5.12)

For example, take this relation xz_{1} = y_{1}y_{4} − y_{2}y_{3} and multiply by y_{2} then replace y_{2}(y_{1}y_{4} − y_{2}y_{3}) by y_{3}(y_{1}y_{3} − y_{2}^{2}) + y_{1}(y_{2}y_{4} − y_{3}^{2}). We end up xz_{1}y_{2} = y_{3}(y_{1}y_{3}−y_{2}^{2})+y_{1}(y_{2}y_{4}−y_{3}^{2}). By using the other two relations y_{1}y_{3}−y_{2}^{2} = −y_{0}y_{4} and xz_{2} = y_{2}y_{4} − y_{3}^{2} of (X, D) we get xz_{1}y_{2} = y_{3}(−y_{0}y_{4}) + y_{1}xz_{2} which gives the desired relation z_{1}y_{2} = y_{1}z_{2} − xy_{3}y_{4}. Similarly we can do the same thing for the second relation in degree 5. Now by using the cubic relation of (X, 2D) together with necessary relations of (X, D) we have so far, we can rewrite f_{2} = x(y_{4}z_{1} − y_{3}z_{2} + xG) which gives another new relation of (X, D) in degree 5

R_{7} : − y_{4}z_{1} + y_{3}z_{2} − xG = 0.  (5.13)

There are three new relations of (X, D) in degree 6

\begin{matrix}R_{6} : & − z_{1}z_{2} + y_{4}^{2}y_{3} − y_{2}G = 0, \\ R_{8} : & − z_{1}^{2} + y_{4}y_{3}^{2} − y_{1}G = 0, \\ R_{9} : & z_{2}^{2} − y_{4}^{3} + y_{3}G = 0 \end{matrix}  (5.14)

coming from multiplying xz_{1} = y_{1}y_{4} − y_{2}y_{3} by either z_{1} or z_{2} and xz_{2} = y_{2}y_{4} − y_{3}^{2} by z_{2}. The equations in (5.11), (5.12), (5.13) and (5.14) give 9 relations of (X, D) in degrees 4^{3}, 5^{3}, 6^{3}.

We claim that there are enough generators. We prove this by contradiction. First, suppose that there is a new generator w in an even degree. Since X(2, 3) is an (X, 2D)-model w is generated by y_{0}, …, y_{4}, a contradiction. Second, assume that there is a new generator w in an odd degree. Then consider xw, which is of even degree. It vanishes on Γ. Therefore

xw = α_{1}(y_{1}y_{3} − y_{2}^{2}) + α_{2}(y_{1}y_{4} − y_{2}y_{3}) + α_{3}(y_{2}y_{4} − y_{3}^{2})

for some α_{i} in k[y_{0}, …, y_{4}] for i = 1, 2, 3. By using the equations in (5.11) w is not a new generator.

We claim that there are enough relations. We divide the proof into two parts. First, assume that there is a new relation f of (X, D) in an even degree. Observe that f is written in the variables y_{0}, …, y_{4}. So set g(y_{0}, …, y_{4}) = f(x, y_{1}, …, z_{1}, z_{2}). Since g is in the ideal of (X, 2D) f is not a new relation. Finally, assume that there is a new relation f in an odd degree. Consider xf in the variables y_{0}, …, y_{4} so that xf is in the ideal I(X, 2D) of (X, 2D). Therefore xf = af_{1} + bf_{2} for some homogeneous polynomials a, b. Since f_{2} = xR_{7} f is written as a combination of f_{1} and R_{7}, a contradiction.

We conclude that (X, D) is a K3 surface with 7A_{1} in ℙ(1, 2, 2, 2, 2, 3, 3) having 9 relations of degrees 4^{3}, 5^{3}, 6^{3} and 16 syzygies of degrees 6^{2}, 7^{6}, 8^{6}, 9^{2}. To see the syzygies, we observe that the relations can be obtained by the submaximal Pfaffians of the following three 5 × 5 skew-symmetric matrices:

(\begin{matrix}0 & x & y_{1} & y_{2} & y_{3} \\ & 0 & y_{2} & y_{3} & y_{4} \\ & & 0 & −xy_{4} & z_{1} \\ & −sym & & 0 & z_{2} \\ & & & & 0 \end{matrix}) , (\begin{matrix}0 & y_{1} & x & −y_{3} & −z_{1} \\ & 0 & y_{2} & z_{1} & y_{3}y_{4} \\ & & 0 & y_{4} & z_{2} \\ & −sym & & 0 & −G \\ & & & & 0 \end{matrix})

and

(\begin{matrix}0 & y_{3} & x & y_{4} & z_{2} \\ & 0 & y_{2} & z_{2} & y_{4}^{2} \\ & & 0 & −y_{3} & −z_{1} \\ & −sym & & 0 & G \\ & & & & 0 \end{matrix}) .

These give 15 syzygies in degrees 6^{2}, 7^{7}, 8^{4}, 9^{2} but one of the syzygies in degree 7 is a combination of others. Moreover, there are two remaining ones in degree 8 which are:

GR_{5} + y_{2}R_{6} − xy_{4}R_{7} + y_{1}R_{9} + z_{2}R_{2} + y_{4}^{2}R_{4} = 0,
GR_{4} − y_{4}R_{8} + z_{1}R_{7} + y_{3}R_{6} = 0.

CHAPTER 5. UNPROJECTION-PROJECTION 83

5.3.1 Lists of possible K3 surfaces in codimension 4

Table 5.2: Weighted K3 surfaces in codimension 4

Nb. Weighted K3 surface Resolution Data
1 X ⊂ ℙ(1, 1, 1, 1, 1, 1, 2)  8 + \frac{1}{2}
2 X ⊂ ℙ(1, 1, 1, 1, 1, 2, 2)  6 + 2 × \frac{1}{2}
3 X ⊂ ℙ(1, 1, 1, 1, 2, 2, 2)  4 + 3×\frac{1}{2}
4 X ⊂ ℙ(1, 1, 1, 1, 2, 3, 4)  4 + \frac{3}{4}
5 X ⊂ ℙ(1, 1, 1, 1, 2, 2, 3)  4 + \frac{1}{2} + \frac{2}{3}
6 X ⊂ ℙ(1, 1, 1, 2, 2, 2, 2)  2 + 4 × \frac{1}{2}
7 X ⊂ ℙ(1, 1, 1, 2, 2, 3, 3)  2 + 2 × \frac{2}{3}
8 X ⊂ ℙ(1, 1, 1, 2, 3, 4, 5)  2 + \frac{4}{5}
9 X ⊂ ℙ(1, 1, 1, 2, 2, 2, 3)  2 + 2 × \frac{1}{2} + \frac{2}{3}
10 X ⊂ ℙ(1, 1, 1, 2, 2, 3, 4)  2 + \frac{1}{2} + \frac{3}{4}
11 X ⊂ ℙ(1, 1, 2, 2, 2, 2, 2)  0 + 6 × \frac{1}{2}
12 X ⊂ ℙ(1, 1, 2, 3, 3, 4, 4)  0 + 2 × \frac{3}{4}
13 X ⊂ ℙ(1, 1, 3, 4, 5, 6, 7)  0 + \frac{6}{7}
14 X ⊂ ℙ(1, 1, 2, 2, 3, 3, 3)  0 + 3 × \frac{2}{3}
15 X ⊂ ℙ(1, 1, 2, 2, 3, 5, 7)  0 + \frac{3·4}{7}
16 X ⊂ ℙ(1, 1, 2, 3, 3, 4, 7)  0 + \frac{2·5}{7}
17 X ⊂ ℙ(1, 1, 2, 2, 2, 3, 4)  0 + 3 × \frac{1}{2} + \frac{3}{4}
18 X ⊂ ℙ(1, 1, 2, 2, 3, 4, 5)  0 + 2 × \frac{1}{2} + \frac{4}{5}
19 X ⊂ ℙ(1, 1, 2, 2, 3, 3, 5)  0 + \frac{2}{3} + \frac{2·3}{5}
20 X ⊂ ℙ(1, 1, 2, 2, 2, 3, 5)  0 + 2 × \frac{1}{2} + \frac{2·3}{5}
21 X ⊂ ℙ(1, 1, 2, 2, 2, 2, 3)  0 + 4 × \frac{1}{2} + \frac{2}{3}

continued on next page

CHAPTER 5. UNPROJECTION-PROJECTION 84

Table 5.2: Weighted K3 surfaces in codimension 4

Nb. Weighted K3 surface Resolution Data
22 X ⊂ ℙ(1, 1, 2, 3, 4, 5, 6)  0 + \frac{1}{2} + \frac{5}{6}
23 X ⊂ ℙ(1, 1, 2, 3, 3, 4, 5)  0 + \frac{2}{3} + \frac{4}{5}
24 X ⊂ ℙ(1, 1, 2, 2, 3, 3, 4)  0 + \frac{1}{2} + \frac{2}{3} + \frac{3}{4}
25 X ⊂ ℙ(1, 2, 2, 2, 2, 3, 3)  −2 + 7 × \frac{1}{2}
26 X ⊂ ℙ(1, 2, 3, 3, 5, 8, 11) −2 + \frac{4·7}{11}
27 X ⊂ ℙ(1, 3, 4, 4, 5, 5, 5)  −2 + 3 × \frac{4}{5}
28 X ⊂ ℙ(1, 3, 4, 4, 5, 9, 13) −2 + \frac{3·10}{13}
29 X ⊂ ℙ(1, 2, 2, 2, 3, 3, 3)  −2 + 4 × \frac{1}{2} + 2 × \frac{2}{3}
30 X ⊂ ℙ(1, 2, 2, 2, 3, 3, 4)  −2 + 5 × \frac{1}{2} + \frac{3}{4}
31 X ⊂ ℙ(1, 2, 2, 2, 3, 3, 5)  −2 + 4 × \frac{1}{2} + \frac{2·3}{5}
32 X ⊂ ℙ(1, 2, 2, 3, 4, 5, 6)  −2 + 4 × \frac{1}{2} + \frac{5}{6}
33 X ⊂ ℙ(1, 2, 2, 3, 3, 4, 4)  −2 + 3 × \frac{1}{2} + 2 × \frac{3}{4}
34 X ⊂ ℙ(1, 2, 2, 3, 3, 3, 3)  −2 + \frac{1}{2} + 4 × \frac{2}{3}
35 X ⊂ ℙ(1, 2, 2, 3, 3, 4, 7)  −2 + 3 × \frac{1}{2} + \frac{2·5}{7}
36 X ⊂ ℙ(1, 2, 2, 3, 3, 5, 8)  −2 + 2 × \frac{1}{2} + \frac{3·5}{8}
37 X ⊂ ℙ(1, 2, 2, 3, 5, 7, 9)  −2 + \frac{1}{2} + \frac{4·5}{9}
38 X ⊂ ℙ(1, 2, 4, 5, 6, 7, 8)  −2 + 3 × \frac{1}{2} + \frac{7}{8}
39 X ⊂ ℙ(1, 2, 3, 3, 3, 4, 5)  −2 + 3 × \frac{2}{3} + \frac{4}{5}
40 X ⊂ ℙ(1, 2, 3, 4, 4, 5, 5)  −2 + 2 × \frac{1}{2} + 2 × \frac{4}{5}
41 X ⊂ ℙ(1, 2, 3, 3, 3, 4, 4)  −2 + 2 × \frac{2}{3} + 2 × \frac{3}{4}
42 X ⊂ ℙ(1, 2, 3, 3, 4, 4, 4)  −2 + \frac{1}{2} + 3 × \frac{3}{4}
43 X ⊂ ℙ(1, 2, 3, 3, 4, 5, 7)  −2 + \frac{2·3}{5} + \frac{2·5}{7}
44 X ⊂ ℙ(1, 2, 3, 4, 4, 5, 9)  −2 + 2 × \frac{1}{2} + \frac{2·7}{9}

continued on next page

CHAPTER 5. UNPROJECTION-PROJECTION 85

Table 5.2: Weighted K3 surfaces in codimension 4

Nb. Weighted K3 surface Resolution Data
45 X ⊂ ℙ(1, 2, 3, 3, 4, 4, 5)  −2 + 2 × \frac{3}{4} + \frac{2·3}{5}
46 X ⊂ ℙ(1, 2, 3, 3, 3, 4, 7)  −2 + 2 × \frac{2}{3} + \frac{2·5}{7}
47 X ⊂ ℙ(1, 2, 3, 4, 5, 5, 7)  −2 + \frac{4}{5} + \frac{3·4}{7}
48 X ⊂ ℙ(1, 2, 3, 3, 4, 5, 8)  −2 + \frac{3}{4} + \frac{3·5}{8}
49 X ⊂ ℙ(1, 3, 5, 6, 7, 8, 9)  −2 + 2 × \frac{2}{3} + \frac{8}{9}
50 X ⊂ ℙ(1, 3, 4, 4, 5, 6, 7)  −2 + 2 × \frac{3}{4} + \frac{6}{7}
51 X ⊂ ℙ(1, 4, 5, 6, 6, 7, 7)  −2 + \frac{1}{2} + 2 × \frac{6}{7}
52 X ⊂ ℙ(1, 3, 4, 5, 5, 6, 6)  −2 + \frac{2}{3} + 2 × \frac{5}{6}
53 X ⊂ ℙ(1, 5, 7, 8, 9, 10, 11) −2 + \frac{2·3}{5} + \frac{10}{11}
54 X ⊂ ℙ(1, 4, 5, 6, 6, 7, 13) −2 + \frac{1}{2} + \frac{2·11}{13}
55 X ⊂ ℙ(1, 3, 4, 5, 5, 6, 11) −2 + \frac{2}{3} + \frac{2·9}{11}
56 X ⊂ ℙ(1, 3, 4, 4, 5, 5, 9)  −2 + \frac{4}{5} + \frac{2·7}{9}
57 X ⊂ ℙ(1, 3, 4, 5, 6, 7, 7)  −2 + \frac{6}{7} + \frac{2·5}{7}
58 X ⊂ ℙ(1, 2, 2, 3, 3, 5, 5)  −2 + \frac{1}{2} + 2 × \frac{2·3}{5}
59 X ⊂ ℙ(1, 2, 3, 3, 4, 7, 10) −2 + \frac{1}{2} + \frac{3·7}{10}
60 X ⊂ ℙ(1, 2, 2, 3, 3, 5, 7)  −2 + \frac{1}{2} + \frac{2}{3} + \frac{3·4}{7}
61 X ⊂ ℙ(1, 6, 8, 9, 10, 11, 12) −2 + \frac{1}{2} + \frac{2}{3} + \frac{11}{12}
62 X ⊂ ℙ(1, 4, 6, 7, 8, 9, 10) −2 + \frac{1}{2} + \frac{3}{4} + \frac{9}{10}
63 X ⊂ ℙ(1, 4, 5, 6, 6, 7, 8)  −2 + \frac{1}{2} + \frac{5}{6} + \frac{7}{8}
64 X ⊂ ℙ(1, 4, 5, 6, 7, 8, 9)  −2 + \frac{1}{2} + \frac{4}{5} + \frac{8}{9}
65 X ⊂ ℙ(1, 3, 4, 5, 6, 7, 8)  −2 + \frac{2}{3} + \frac{3}{4} + \frac{7}{8}
66 X ⊂ ℙ(1, 3, 4, 5, 5, 6, 7)  −2 + \frac{2}{3} + \frac{4}{5} + \frac{6}{7}
67 X ⊂ ℙ(1, 2, 3, 3, 4, 5, 6)  −2 + \frac{1}{2} + 2 × \frac{2}{3} + \frac{5}{6}

continued on next page

CHAPTER 5. UNPROJECTION-PROJECTION 86

Table 5.2: Weighted K3 surfaces in codimension 4

Nb. Weighted K3 surface Resolution Data
68 X ⊂ ℙ(1, 2, 3, 4, 5, 6, 7)  −2 + 2 × \frac{1}{2} + \frac{2}{3} + \frac{6}{7}
69 X ⊂ ℙ(1, 2, 3, 4, 4, 5, 6)  −2 + 2 × \frac{1}{2} + \frac{3}{4} + \frac{5}{6}
70 X ⊂ ℙ(1, 2, 3, 4, 5, 5, 6)  −2 + \frac{1}{2} + \frac{2·3}{5} + \frac{5}{6}
71 X ⊂ ℙ(1, 2, 3, 3, 4, 5, 5)  −2 + \frac{2}{3} + \frac{4}{5} + \frac{2·3}{5}
72 X ⊂ ℙ(1, 2, 3, 3, 4, 4, 7)  −2 + \frac{1}{2} + \frac{3}{4} + \frac{2·5}{7}
73 X ⊂ ℙ(1, 2, 2, 3, 3, 3, 4)  −2 + 2 × \frac{1}{2} + 2 × \frac{2}{3} + \frac{3}{4}
74 X ⊂ ℙ(1, 2, 2, 3, 3, 4, 5)  −2 + 3×\frac{1}{2} + \frac{2}{3} + \frac{4}{5}
75 X ⊂ ℙ(1, 2, 2, 3, 3, 4, 5)  −2 + 2 × \frac{1}{2} + \frac{3}{4} + \frac{2·3}{5}
76 X ⊂ ℙ(1, 2, 2, 3, 3, 5, 7)  −2 + \frac{1}{2} + \frac{2}{3} + \frac{3·4}{7}
77 X ⊂ ℙ(1, 2, 2, 3, 3, 3, 5)  −2 + \frac{1}{2} + 2 × \frac{2}{3} + \frac{2·3}{5}
78 X ⊂ ℙ(1, 2, 3, 3, 4, 4, 5)  −2 + \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5}
79 X ⊂ ℙ(2, 3, 5, 5, 7, 12, 17) −4 + \frac{7·10}{17}
80 X ⊂ ℙ(2, 2, 3, 5, 7, 9, 11) −4 + 3 × \frac{1}{2} + \frac{5·6}{11}
81 X ⊂ ℙ(2, 3, 4, 5, 7, 7, 11) −4 + \frac{2·5}{7} + \frac{5·6}{11}
82 X ⊂ ℙ(3, 3, 4, 5, 7, 10, 13) −4 + 2 × \frac{2}{3} + \frac{4·9}{13}
83 X ⊂ ℙ(3, 5, 6, 7, 8, 11, 14) −4 + \frac{5}{6} + \frac{5·9}{14}
84 X ⊂ ℙ(5, 5, 6, 7, 8, 9, 11) −4 + 2 × \frac{2·3}{5} + \frac{2·9}{11}
85 X ⊂ ℙ(2, 3, 4, 4, 5, 7, 11) −4 + 4 × \frac{1}{2} + \frac{3·8}{11}
86 X ⊂ ℙ(2, 3, 3, 4, 5, 7, 9)  −4 + 3 × \frac{2}{3} + \frac{4·5}{9}
87 X ⊂ ℙ(2, 3, 3, 4, 5, 5, 5)  −4 + \frac{2}{3} + 3 × \frac{2·3}{5}
88 X ⊂ ℙ(2, 3, 4, 5, 5, 7, 7)  −4 + \frac{3}{4} + 2 × \frac{3·4}{7}
89 X ⊂ ℙ(2, 5, 6, 7, 9, 11, 13) −4 + \frac{5}{6} + \frac{6·7}{13}
90 X ⊂ ℙ(2, 2, 3, 3, 4, 5, 5)  −4 + 4 × \frac{1}{2} + 2 × \frac{2·3}{5}

continued on next page

CHAPTER 5. UNPROJECTION-PROJECTION 87

Table 5.2: Weighted K3 surfaces in codimension 4

Nb. Weighted K3 surface Resolution Data
91 X ⊂ ℙ(2, 2, 3, 3, 4, 5, 7)  −4 + 4 × \frac{1}{2} + \frac{2}{3} + \frac{3·4}{7}
92 X ⊂ ℙ(2, 3, 3, 4, 4, 5, 7)  −4 + 3 × \frac{1}{2} + 2 × \frac{2}{3} + \frac{2·5}{7}
93 X ⊂ ℙ(2, 3, 3, 4, 5, 8, 11) −4 + 2 × \frac{1}{2} + \frac{2}{3} + \frac{4·7}{11}
94 X ⊂ ℙ(2, 3, 4, 5, 5, 7, 12) −4 + \frac{1}{2} + \frac{3}{4} + \frac{5·7}{12}
95 X ⊂ ℙ(2, 3, 4, 5, 5, 5, 6)  −4 + 2 × \frac{1}{2} + \frac{4}{5} + 2 × \frac{2·3}{5}
96 X ⊂ ℙ(2, 3, 4, 5, 5, 6, 8)  −4 + 3 × \frac{1}{2} + \frac{4}{5} + \frac{3·5}{8}
97 X ⊂ ℙ(2, 3, 4, 5, 5, 7, 9)  −4 + \frac{3}{4} + \frac{2·3}{5} + \frac{4·5}{9}
98 X ⊂ ℙ(3, 4, 4, 5, 6, 7, 10) −4 + \frac{1}{2} + 2 × \frac{3}{4} + \frac{3·7}{10}
99 X ⊂ ℙ(2, 3, 4, 4, 5, 5, 7)  −4 + 2 × \frac{1}{2} + 2 × \frac{3}{4} + \frac{3·4}{7}
100 X ⊂ ℙ(2, 5, 6, 7, 7, 8, 9)  −4 + 2 × \frac{1}{2} + \frac{6}{7} + \frac{4·5}{9}
101 X ⊂ ℙ(2, 3, 4, 5, 7, 7, 9)  −4 + \frac{1}{2} + \frac{2·5}{7} + \frac{4·5}{9}
102 X ⊂ ℙ(2, 3, 4, 5, 7, 9, 11) −4 + \frac{2}{3} + \frac{3}{4} + \frac{5·6}{11}
103 X ⊂ ℙ(2, 3, 3, 4, 5, 5, 7)  −4 + 2 × \frac{2}{3} + \frac{2·3}{5} + \frac{3·4}{7}
104 X ⊂ ℙ(3, 4, 5, 6, 7, 10, 13) −4 + \frac{1}{2} + \frac{4}{5} + \frac{4·9}{13}
105 X ⊂ ℙ(3, 4, 5, 5, 6, 7, 9)  −4 + 2 × \frac{2}{3} + \frac{2·3}{5} + \frac{2·7}{9}
106 X ⊂ ℙ(4, 4, 5, 6, 7, 9, 13) −4 + 2 × \frac{1}{2} + \frac{3}{4} + \frac{3·10}{13}
107 X ⊂ ℙ(3, 4, 5, 5, 6, 7, 9)  −4 + 2 × \frac{2}{3} + \frac{2·3}{5} + \frac{2·7}{9}
108 X ⊂ ℙ(3, 4, 4, 5, 6, 7, 7)  −4 + \frac{1}{2} + \frac{3}{4} + 2 × \frac{2·5}{7}
109 X ⊂ ℙ(3, 4, 4, 5, 6, 7, 11) −4 + \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{3·8}{11}
110 X ⊂ ℙ(3, 4, 5, 5, 6, 7, 8)  −4 + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \frac{3·5}{8}
111 X ⊂ ℙ(4, 5, 5, 6, 7, 8, 9)  −4 + \frac{1}{2} + \frac{4}{5} + \frac{2·3}{5} + \frac{2·7}{9}
112 X ⊂ ℙ(3, 4, 5, 5, 6, 7, 7)  −4 + \frac{2}{3} + \frac{4}{5} + \frac{2·3}{5} + \frac{2·5}{7}
113 X ⊂ ℙ(2, 3, 3, 4, 5, 5, 8)  −4 + \frac{1}{2} + \frac{2}{3} + \frac{2·3}{5} + \frac{3·5}{8}

continued on next page

Table 5.2: Weighted K3 surfaces in codimension 4

Nb.  Weighted K3 surface  Resolution Data
114 X ⊂ ℙ(2, 3, 4, 5, 5, 6, 7)  −4 + 2 × \frac{1}{2} + \frac{2}{3} + \frac{4}{5} + \frac{3·4}{7}
115 X ⊂ ℙ(2, 3, 3, 4, 4, 5, 5)  −4 + 2 × \frac{1}{2} + 2 × \frac{2}{3} + \frac{3}{4} + \frac{2·3}{5}

Examples in codim 4, except #25, #84, #111, in Table 5.2 can be obtained by unprojecting from successive constructions of K3 surfaces in codimensions 2, 3 as in Example 5.11, and #84, #111 can only be constructed by unprojecting from #61, #65 in the list of codim 3 respectively. On the other hand, to check quasismooth there is some work that needs to be done. For #25, see Example 5.14.

Additionally, there are twenty three candidates for K3 surfaces in codim 4 which we do not understand well how to get from unprojection or something else.

Table 5.3: Candidates for weighted K3 surfaces in codimension 4

Nb. Weighted K3 surface  Formal Resolution Data
1 X ⊂ ℙ(2, 2, 3, 4, 5, 5, 6) −4 + 7 × \frac{1}{2} + \frac{4}{5}
2 X ⊂ ℙ(3, 3, 4, 4, 5, 5, 6) −4 + 4 × \frac{2}{3} + 2 × \frac{3}{4}
3 X ⊂ ℙ(3, 3, 4, 5, 5, 6, 7) −4 + 5 × \frac{2}{3} + \frac{4}{5}
4 X ⊂ ℙ(2, 3, 3, 3, 4, 4, 5) −4 + 2 × \frac{1}{2} + 5 × \frac{2}{3}
5 X ⊂ ℙ(2, 2, 3, 3, 3, 4, 4) −4 + 6 × \frac{1}{2} + 2 × \frac{2}{3}
6 X ⊂ ℙ(2, 2, 2, 3, 3, 3, 4) −4 + 8 × \frac{1}{2} + \frac{2}{3}
7 X ⊂ ℙ(2, 3, 4, 4, 4, 5, 5) −4 + 4 × \frac{1}{2} + 3 × \frac{3}{4}
8 X ⊂ ℙ(2, 5, 6, 7, 8, 8, 9) −4 + 4 × \frac{1}{2} + \frac{2·3}{5} + \frac{7}{8}
9 X ⊂ ℙ(2, 2, 3, 3, 4, 4, 5) −4 + 6 × \frac{1}{2} + \frac{2}{3} + \frac{3}{4}
10 X ⊂ ℙ(2, 6, 7, 8, 9, 9, 10) −4 + 5 × \frac{1}{2} + \frac{2}{3} + \frac{8}{9}

Table 5.3: Candidates for weighted K3 surfaces in codimension 4

Nb. Weighted K3 surface  Formal Resolution Data
11 X ⊂ ℙ(2, 4, 5, 6, 7, 7, 8) −4 + 5 × \frac{1}{2} + \frac{3}{4} + \frac{6}{7}
12 X ⊂ ℙ(2, 4, 5, 5, 6, 6, 7) −4 + 5 × \frac{1}{2} + \frac{4}{5} + \frac{5}{6}
13 X ⊂ ℙ(4, 5, 6, 7, 7, 8, 9) −4 + 2 × \frac{3}{4} + \frac{5}{6} + \frac{3·4}{7}
14 X ⊂ ℙ(3, 5, 6, 7, 7, 8, 9) −4 + 3 × \frac{2}{3} + \frac{2·3}{5} + \frac{6}{7}
15 X ⊂ ℙ(3, 4, 4, 5, 5, 6, 7) −4 + \frac{2}{3} + 3 × \frac{3}{4} + \frac{2·3}{5}
16 X ⊂ ℙ(3, 6, 7, 8, 8, 9, 10) −4 + \frac{1}{2} + 4 × \frac{2}{3} + \frac{7}{8}
17 X ⊂ ℙ(2, 3, 4, 5, 6, 6, 7) −4 + 4 × \frac{1}{2} + 2 × \frac{2}{3} + \frac{5}{6}
18 X ⊂ ℙ(4, 6, 7, 7, 8, 9, 10) −4 + 2 × \frac{1}{2} + \frac{3}{4} + \frac{6}{7} + \frac{2·5}{7}
19 X ⊂ ℙ(6, 6, 7, 8, 9, 10, 11) −4 + 2 × \frac{1}{2} + 2 × \frac{2}{3} + \frac{5}{6} + \frac{6}{7}
20 X ⊂ ℙ(3, 4, 5, 6, 6, 7, 8) −4 + \frac{1}{2} + 3 × \frac{2}{3} + \frac{3}{4} + \frac{4}{5}
21 X ⊂ ℙ(2, 3, 4, 4, 5, 5, 6) −4 + 4 × \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5}
22 X ⊂ ℙ(5, 6, 6, 7, 8, 9, 10) −4 + \frac{1}{2} + \frac{2}{3} + \frac{2·3}{5} + 2 × \frac{5}{6}
23 X ⊂ ℙ(4, 5, 6, 6, 7, 8, 9) −4 + 2 × \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \frac{4}{5} + \frac{5}{6}

Appendix

In this appendix we describe a construction of a quartic hypersurface Y = X(4) ⊂ ℙ^{3} which touches a plane along a conic C and in general has six nodes. Then we show that this leads to a K3 surface X(4, 6) ⊂ ℙ(1, 2, 2, 2, 3) with six nodes which is a (Y, C)-model. This means that X(4, 6) = Proj R(Y, C) where R(Y, C) = ⨁_{n≥0} H^{0}(Y, O_{Y} (nC)).

Let y_{1}, y_{2}, y_{3} be homogeneous coordinates on ℙ^{2}. The cone C′ is given by the zero locus of f = y_{1}y_{3} − y_{2}^{2}. Now define a quartic hypersurface Y

(f^{2} + y_{0}g = 0) ⊂ ℙ^{3} = ℙ(y_{0}, …, y_{3})

where g is a general homogeneous polynomial of degree 3 in k[y_{0}, …, y_{3}]. It is easy to see that Y is nonsingular away from C: y_{0} = f = 0 by Bertini’s theorem and in general has six nodes along C. The hyperplane section (y_{0} = 0) on Y is Y ∩ (y_{0} = 0) = 2C satisfying (2C)^{2} = 4. So Y is a (Y, 2C)-model. By the Riemann–Roch formula on a singular surface we have h^{0}(Y, C) = 1. This implies that there exists a generator x of degree 1 such that y_{0} = x^{2}. Since h^{0}(Y, 4C − C) = h^{0}(Y, 3C) = 5 there is a generator z of degree 3 such that xz = f, which gives a relation of degree 4. By substituting this relation into the defining equation of Y we get another relation, namely (z^{2} + g = 0), of degree 6. Define X(4, 6) by

(xz − f = z^{2} + g = 0) ⊂ ℙ(1, 2, 2, 2, 3).

One can easily show that we do not get any new generators and relations by inspecting H^{0}(Y, nC) for n > 3 (or use the same strategy as in Example 5.14 assuming that there are new generators or relations in order to get a contradiction). Therefore X(4, 6) is a (Y, C)-model.

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