MA4L7 Algebraic Curves

Lecturer: Miles Reid

Term(s): Term 2

Status for Mathematics students: List C

Commitment: 30 Lectures

Assessment: 85% exam, 15% assessed worksheets

Formal registration prerequisites: None

Assumed knowledge: Some familiarity with basic ideas of commutative algebra is a prerequisite. As a rough guide, the lectures need the first half of MA3G6 Commutative Algebra. More specifically, the main technical items are localisation (partial rings of fraction of an integral domain), local rings. The definitions and ideas from the first half of MA4A5 Algebraic Geometry are also prerequisites.

Useful background: Material such as field extensions and ideals in the polynomial ring from MA3D5 Galois Theory will serve as useful motivation. The idea of integral elements of a number field MA3A6 Algebraic Number Theory is a good warm-up for integral closure that will be used in the course. The idea of meromorphic function from MA3B8 Complex Analysis will be mentioned in explaining the purely algebraic discussion of zeros and poles of a rational function. In a similar way, the Cauchy integral theorem is good motivation for the full statement of the Riemann-Roch theorem, although it is not needed for the proof.

Synergies: The course is a basic introduction to the study of algebraic varieties (and schemes) and their cohomology. The Riemann-Roch theorem for curves is a first major step towards the classification of algebraic curves, surfaces and higher dimensional varieties, that makes up a large component of modern algebraic geometry, and has applications across the mathematical sciences and theoretical physics.

Leads to: The following modules have this module listed as assumed knowledge or useful background:

Content: The module covers basic questions on algebraic curves. The first sections establishes the class of non-singular projective algebraic curves in algebraic geometry as an object of study, and for comparison and motivation, the parallel world of compact Riemann surfaces. After these preliminaries, most of the rest of the course focuses on the Riemann-Roch space $\mathcal{L}(C,D)$, the vector space of meromorphic functions on a compact Riemann surface or a non-singular projective algebraic curve with poles bounded by a divisor $D$ - roughly speaking, allowing more poles gives more meromorphic functions.

The statement of the Riemann-Roch theorem

$\dim\mathcal{L}(C,D) \ge 1-g+\deg D.$

It comes with sufficient conditions for equality. The main thrust of the result is to provide rational functions that allows us to embed $C$ into projective space $\mathbb{P}^n$. The formula involves an invariant called the genus $g(C)$ of the curve. In intuitive topological terms, we think of it as the ''number of holes''. However, it has many quite different characterisations in analysis and in algebraic geometry, and is calculated in many different ways. The logical relations between these treatments is a little complicated. A middle section of the course emphasizes the meaning and purpose of the theorem (independent of its proof), and give important examples of its applications.

The proof of RR is based on commutative algebra. Algebraic varieties have many different types of rings associated with them, including affine coordinate rings, homogeneous coordinate rings, their integral closures, and their localisations such as the DVRs that correspond to points of a non-singular curve. Footnote to the course notes include (as non-examinable material) references to high-brow ideas such as coherent sheaves and their cohomology and Serre--Grothendieck duality.

Learning Outcomes:

By the end of the module the student should be able to:

• Demonstrate understanding of the basic concepts, theorems and calculations related to projective curves defined by homogeneous polynomials of low degree
• Demonstrate understanding of the basic concepts, theorems and calculations that relate the zeroes and poles of rational functions with the general theory of discrete valuation rings and divisors on projective curves
• Demonstrate knowledge and understanding of the statement of the Riemann-Roch theorem and an understanding of some of its applications
• Demonstrate understanding of the proof of the Riemann-Roch theorem

Books:

Frances Kirwan, Complex Algebraic Curves, LMS student notes

William Fulton, Algebraic Curves: An Introduction to Algebraic Geometry online at www.math.lsa.umich.edu/~wfulton/CurveBook.pdf

I.R. Shafarevich, Basic Algebraic Geometry (especially Part 1, Chapter 3, Section 3.7)

Robin Hartshorne, Algebraic Geometry, (Chapter 4 only)

The lecturer's notes will be made available during the course.