Geometry of Algebraic Curves
Course Description: The aim of this course is to develop and explore the theory of algebraic curves from the viewpoint of modern algebraic geometry. The subject matter combines tools from commutative and homological algebra, Galois theory and algebraic number theory, among others. One of the main objectives will be to justify the use of "geometry" in the course's title as every effort will be made to accompany the algebraic formalism with geometric intuition.
Coordinates: Tuesdays 10-11 & Wednesdays 10-12 @ Frank Adams 1.
Bibliography:
Frances Kirwan: Complex Algebraic Curves, London Mathematical Society Student Texts, 23, Cambridge University Press, 1992.
William Fulton: Algebraic Curves. An Introduction to Algebraic Geometry, Reprint of 1969 original, Addison-Wesley, 1989.
Nigel Hitchin, Dominic Joyce: Lecture Notes on Algebraic Curves
Tuesday 8 November:
Lecture Log: Introduction to Algebraic Geometry, Motivation, Woffle & Omnipresence.
Assessment: None.
Wednesday 9 November:
Lecture Log :
Projective spaces. Points and lines in the projective plane, projective transformations, Möbius transformations, Points in general position.
Conics. Symmetric bilinear forms, Quadtratic forms affine charts, rational parametrization.
Assessment:
Hitchin-Joyce Chapters 1 &2.
Tuesday 15 November:
Lecture Log: More on rational parametrization of conics. Application to Pythagorean triples. Introduction to resultants
Assessment: None.
Wednesday 16 November:
Lecture Log :
Resultants. Two curves intersect in at least one point and at most mn points. Non-singularity implies irreducibility. Irreducibility implies finiteness of singular locus.
Intersection multiplicities. Bézout's theorem. Singular points have intersection multiplicity greater than one. Non-singular curves intersect with their tangents with multiplicity greater than one. Multiplicity equal to one is equivalent to tangent lines being distinct.
Assessment:
Kirwan Sections 3.1 and 3.2. Make sure you study the proofs sketched in class.
Monday 21 November:
Lecture Log:
Riemann surfaces and examples: The projective line/Riemann sphere, the torus, nonsingular curves as Riemann surfaces.
Holomorphic maps between Riemann surfaces, Meromorphic functions, introduction to branched covers.
Assessment: Hitchin-Joyce section 4.1, Kirwan section 5.2.
Tuesday 22 November:
Lecture Log :
Meromorphic functions via poles.
More on branched covers. Fibers of points, unramified points, branch points and ramification points. The ramification index.
Topological properties. Compactness and the Hausdorff property for projective spaces and projective plane curves
Assessment:
Kirwan sections 2.2 & 4.2, Hitchin-Joyce section 4.2.
Monday 28 November:
Lecture Log:
Triangulations and Euler characteristics of Riemann surfaces.
Equivalent definitions of the genus of a non-singular curve. The Riemann-Hurwitz and the genus-degree formulas.
Tori & Cubic curves, the Weierstrass p-function.
Differentiation of meromorphic functions on Riemann surfaces
Assessment: Kirwan sections 4.3, 5.1, 6.1. Hitchin-Joyce sections 4.3, 4.4.
Tuesday 29 November:
Lecture Log :
Meromorphic differentials: two equivalent definitions, integration, recovering the lattice corresponding to a cubic, Abel's Theorem.
Divisors: definition, basic properties. The degree of a divisor, effective and principal divisors, the canonical divisor class.
Riemann-Roch spaces. Definition, intuition and basic properties.
Assessment:
Hitchin-Joyce sections 5.1 & 5.2. Kirwan sections 6.1 & 6.3.
Friday 2 December:
Problem session.
Monday 5 December:
Lecture Log:
Riemann-Roch spaces. Proof of fundamental properties.
The RR theorem for divisors of intersection with lines.
Corollaries of the RR: meromorphic=rational, genus as the dimension of holomorphic differentials
Proof of Riemann's inequality.
Assessment: Hitchin-Joyce sections 5.3 & 5.4. Kirwan section 6.3.
Tuesday 6 December:
Lecture Log :
Recap of proof of Riemann's inequality.
Proof of the "Roch" part of RR.
Proof of pending lemmata.
Holomorphic differentials on algebraic curves
Assessment:
Hitchin-Joyce sections 5.3 & 5.4. Kirwan section 6.3.
Friday 9 December:
Problem session.