# MA4L7 Algebraic curves

The Warwick UG Handbook entry is here. I teach the module in 2021-22,
Term 2. The material and the exam questions may differ from those
of the module taught in Term 1 of 2020-21 by Prof. Diane Maclagan.

### Lectured material

The lectures are in B3.02 and on Teams at the following times:
Mon 10:00, Thu 9:00 and Fri 9:00.
Videos of the lectures and some plain text descriptions are in my
DropBox folder.

### Lecture Notes

I hope to consolidate my lecture notes into a textbook.
The first introductory section Chapter 0 should consist
of independent colloquial sections not involving too many
prerequisites, based on the first 4 lectures, together with
the first lecture of 2020 as a detailed abstract and
description of the course contents.

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

The following is a record of the lecture course as I gave it in 2020. The final
Part 5 is a somewhat novel approach to constructing the canonical class K_C
to complete the proof of RR that seems simpler and more convincing than
treatments in the current literature.

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.

Example Sheet 1
Example Sheet 2
Example Sheet 3
Example Sheet 4
Example Sheet 5
Jun 2020 exam

### Scrap

The first lecture tries to outline in approachable colloquial terms
the idea that the course contents is easy, but built on
sophisticated and sometimes difficult prerequisites from
several areas.

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.