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.

Mon 10:00, Thu 9:00 and Fri 9:00.

Videos of the lectures and some plain text descriptions are in my

The first introductory section

of independent colloquial sections not involving too many

prerequisites, based on the

the

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.)

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.

Part 1

Part 2

Part 3

Castenuovo free pencil trick

Max Noether's theorem

Linearly general position

Part 4

Part 5

Example Sheet 2

Example Sheet 3

Example Sheet 4

Example Sheet 5

Jun 2020 exam

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.