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MA4L7 Content

Content:

The module covers basic questions on algebraic curves. The first sections establishes the class of nonsingular 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 nonsingular 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 nonsingular curve. Footnote to the course notes include (as nonexaminable 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.