MATH 7670 - Algebraic Geometry

Michael Stillman, spring 2014

Topic: Coherent sheaf cohomology, its computation, and applications to projective algebraic geometry

Sheaves and their cohomology are used in many places in algebraic geometry, and they form a powerful tool, both for theoretical advances, as well as for understanding concrete varieties and examples.

In this course, we will introduce sheaves, and cohomology of sheaves on affine and projective varieties, in a manner which will be full of examples relating these abstract ideas with classical geometry. We will also show how to compute with these concepts in concrete situations. We will be working towards some specific applications of cohomology in algebraic geometry, for example Riemann-Roch for curves (and applications of that), invariants of projective varieties and/or rationality of surfaces.

Rough outline

(Some other topics will be included as we go along, e.g. regularity of sheaves, blowups). The exact list of topics is subject to change!

  1. Sheaves and basic properties (e.g. stalks, exact sequences)
  2. Coherent sheaves
  3. Key Example: Cartier divisors and invertible sheaves.
  4. Structure of coherent sheaves on projective varieties
  5. Sheaf cohomology via Cech cohomology
  6. Ext and Koszul complexes
  7. Computing sheaf cohomology
  8. Local duality
  9. Key Example: Sheaf of differentials, canonical sheaf, normal sheaf.
  10. Application: Invariants: Euler characteristic, Hodge diamond
  11. Serre duality
  12. Riemann-Roch for curves
  13. Applications of Riemann-Roch
  14. maps to projective space and linear systems
  15. Applications to geometry of curves on a surface

Algebraic geometry is a subject where you must "learn by doing". I will assign regular homework, and we will discuss the solutions during class.

We will assume a small number of clearly stated results, as their proof sometimes takes us too far afield, and instead we will concentrate on the use of these results.

Prerequisites: Some exposure to basic algebraic geometry and commutative algebra, including affine and projective varieties, regular and rational maps, and localization, free resolutions, Groebner bases, at the level of Peeva's course Math 6340 in Fall 2013.