MATH 767: Introduction to Algebraic Geometry, part II (Fall 2007)

Instructor: Michael Stillman

Meeting Time & Room

Prerequisite: A previous elementary course in algebraic geometry, as in chapter 1 of Hartshorne, or Chapters 1 and 2 of Shafarevich. We will review what we need to assume about divisors.

Textbooks: Shafarevich, volumes I and II

This semester, we will start with divisors and differentials (Chapter 3 in Shafarevich), introducing sheaves very quickly. Our plan is to cover chapters 3 thru 6. We will cover sheaf cohomology as well, something that Shafarevich surprisingly omits.

Throughout the semester we will stress examples.

Tentative list of topics:

Chapter 3. Divisors

  • Very quick review of chapter III.1 Divisors
  • Divisors on curves
  • Basic properties
  • Example: Plane cubic curves
  • Differential forms
    • Canonical divisor
    • Riemann-Roch formula (we will only prove part of it, but examine consequences of the stronger statement)
    • Riemann-Hurwitz formula

Chapter 4. Intersection numbers

This chapter contains many of the important basic results about algebraic surfaces and curves on them.

  • Construction and basic (important!) properties of intersection numbers
  • Bezout's theorem
  • Genus formula for a curve on a surface
  • Riemann-Roch for surfaces
  • The smooth cubic surface and the lines on it
  • Relationship of blowups to intersection numbers for surfaces
  • Surface singularities

Chapter 5. Sheaves and schemes

We will probably have defined and used sheaves earlier than this, but here will use them to help define the notion of scheme.

  • Spectrum of a ring
  • Sheaves
  • Schemes (including a geometric feeling for what they are)
  • Products of schemes

Chapter 6. Varieties, vector bundles and coherent sheaves

Here we restrict to the schemes that an algebraic geometer uses days to day. The key constructions are coherent sheaves and vector bundles. These turn out to be very closely related to divisors, and to everything else we have covered.

There are many cool results in this chapter. We probably will have to pick and choose which ones we we wish to study.

Chapter N. Cohomology of coherent sheaves

We show how to compute cohomology, giving examples and applications. This is crucial to understand modern algebraic geometry, but for some reason, Shafarevich leaves this topic out. We put it back in.

Last modified:August 14, 2007