MA4L7 Algebraic curves

The Warwick UG Handbook entry is here. I teach the module in 2025-26,
Term 2. The material and the exam questions may differ from those
of the module taught in 2024-25 by Rob Silversmith

Timetable

MA4L7 Algebraic curves
Mon 14:00 in B1.01 from Mon 12th Jan 2026
Tue  9:00 in B3.02
Fri 15:00 in B3.01

Example classes with Marc Truter 	
Thu 16:00 in B3.01

Pre-course worksheets

Worksheet 2 Compact Rieman surfaces

Worksheet 3 Homogeneous forms and rational functions on PP^1

First 2026 lecture notes and worksheets

I may find time to beat this up a bit, although it is more useful to
spend time on future lectures. The material will migrate to the Moodle
page later in Week 2, together with details of assessed work
and deadlines.

This is an advanced course for 4th year MMath, MSc and starting
PhD students, I assume results from previous modules on complex
analysis, geometry and topology, algebra of field extensions and
commutative algebra of polynomial rings -- at least the statements
and how to apply them in practice. The course proper is in algebraic
geometry, and I will treat the rigorous material I need either in the
lectures or with detailed references.

I offer the above three preliminary worksheets on the prerequisites --
please skim through this stuff, or look it up on Wikipedia, or study it
more seriously as needed. It is not part of the course from the point
of view of rigour or logical argument, but rigour is not much use if
you don't know what is going on. At some point you have to learn
to juggle with more than one ball, and the background material
should give some motivation to help you in this respect. You won't
get very far if you don't have the material somewhere in your mind.

Lectured material

Some videos of the lectures from 2021-2022 together
with some plain text descriptions are in my DropBox folder.

Lecture notes from 2022

Chapter 1 establishes nonsingular projective curves as an
object of study.

Chapter 2 introduces the RR spaces L(C,D) of a divisor D
on a nonsingular projective curve C, and proves the RR
theorem modulo three Main Propositions I-II-III that are
deferred to Chapter 4.

Chapter 3 discusses several of the standard applications of RR:
very ample divisor, the dichotomy between canonical embedding
and hyperelliptic, the multiplication map of RR spaces and its
applications to Clifford's theorem and the Castelnuovo free
pencil trick. I discuss the theorem on linear general position,
and I attach Samuel's proof of the nonexistence of "strange" curves.

Chapter 4A introduces some of the ideas and methods of
graded rings and defines the "sections ring" R(C,D). It proves Main
Proposition I and Proposition II in a coarse form.
(The Chap. 4A/4B division is a temporary device for the 2022 notes.)

Chapter 4B introduces the canonical module K(C,D) dual to R(C,D),
and discusses how the ideas of graded rings and their modules
give a complete treatment of the numerology of RR for multiples of D.
Explaining how the canonical module is realised as the "sections module"
of a divisor KC is a slightly trickier issue. It gives the proof of
Main Proposition III, so the full RR theorem. My approach is somewhat
novel, since it introduces the canonical class KC without any mention
of Kaehler differential 1-forms.

As an addendum I treat standard results such as the ramification divisor
of a separable map and Hurwitz's theorem. Finally, I explain how the
canonical class relates to 1-forms. (Some of this remains provisional.)

Notes from 2020

I have not yet had time to polish this up adequately, and some parts obviously
need more work to bring them up to textbook standard. I hope to return to this at
some future point.

Part 1
Part 2
Part 3
 Castenuovo free pencil trick
 Max Noether's theorem
 Linearly general position
Part 4
Part 5

Worksheets

Scrap

Normal characterises DVRs A brief self-contained treatment of a
key result on nonsingularity.

Here is the old directory containing the 2019 notes and
worksheets and other scrap.