The Warwick UG Handbook entry is
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.
establishes nonsingular projective curves as an
object of study.
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.
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
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.)
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.
Castenuovo free pencil trick
Max Noether's theorem
Linearly general position
Normal characterises DVRs A brief self-contained treatment of a
key result on nonsingularity.
Here is the
directory containing the 2019 notes and
worksheets and other scrap.