Topics in Algebraic Geometry

Tim Dokchitser, TCC, Autumn 2020

Lecture times
Thursdays 2-4pm, 15 October - 3 December, on MS Teams.

Lecture 1. Affine varieties and affine algebraic groups (Oct 15).
Lecture 2. Complete varieties and curves (Oct 22).
Lecture 3. Local rings, genus and differentials on curves (Oct 29).
Lecture 4. Riemann-Roch and models of curves (Nov 5).
Lecture 5. Picard groups of curves (Nov 12).
Lecture 4. Abelian varieties (Nov 19).
Lecture 7. Twists and forms (Nov 26).
Lecture 8. Moduli problems (Dec 3).

Assumed knowledge
I will assume some familiarity with algebraic varieties, and especially with algebraic curves. I will start with a brief review in the first lectures, but basically I will assume that you have seen Zariski topology, regular and rational maps, projective space, non-singular curves, divisors and Riemann-Roch. I will probably manage to stick to varieties throughout the course and avoid scheme-theoretic language completely.

The course will focus on curves and their Jacobians. The first half will discuss them and a few background topics: algebraic groups, abelian varieties, divisors, differential and genus of curves. After that (if there is time!), I will either go into the direction of functors and moduli spaces, or discuss regular and semistable models of curves, depending on the interest.

Lecture notes
There is a (constantly updated) informal set of lecture notes for the course.

Through 4 homework problems, one a fortnight. Please email your solutions to
Homework 1, to be handed in by 29th of October
Homework 2, to be handed in by 12th of November
Homework 3, to be handed in by 26th of November
Homework 4, to be handed in by 10th of December