MATH 7670: An Intermediate Course in Algebraic Geometry: Curves and Surfaces (Spring 2012)

Instructor: Michael Stillman

We will cover the material in Chapters 4 and 5 of Hartshorne, although, for some topics, we will use a more modern approach. We do not assume Chapter 3 (Cohomology of sheaves), instead we introduce these techniques in the context of using them to study algebraic curves and surfaces.

A tentative list of topics:

1. Algebraic Curves (Hartshorne, Chapter 4)
   1. Working person's guide to sheaf cohomology (part I, other parts covered later)
   2. Divisors and line bundles on curves
   3. Riemann-Roch for curves
   4. Hurwitz theorem
   5. embeddings in projective space
   6. elliptic curves
   7. the canonical embedding, and Clifford's theorem
   8. classification of smooth space curves
2. Projective algebraic surfaces (Hartshorne, Chapter 5)
   1. adjunction, intersection theory on surfaces, Riemann-Roch for surfaces numerical invariants.
   2. blowups
   3. ruled surfaces and vector bundles
   4. cubic surface
   5. birational geometry of surfaces (done with a more modern approach)
   6. birational classification of surfaces

The specific topics on which we focus, and how fast we cover the material, will depend on the interests and background of the participants.

These chapters in Hartshorne are very nicely written, contain beautiful geometry and examples, and also allow one to apply and really internalize the abstract machinery of sheaves and sheaf cohomology, enabling it as a useful tool. The exercises in Hartshorne are also excellent.

In order to learn algebraic geometry, it is important to do it, not just read it or listen to it. So, we will do many examples and exercises.

Textbook: Hartshorne: Algebraic Geometry.

Prerequisites: algebraic geometry at the level of chapters 1, 2 of Hartshorne. At a minimum: you just need the language of sheaves and schemes, and a previous course in algebraic geometry (as long as you don't mind believing some results, or looking them up on your own).