Cornell Math - MATH 767, Spring 2007

MATH 767: Algebraic Geometry (Spring 2007)

Instructor: Michael Stillman

Meeting Time & Room

Prerequisites: A course in commutative algebra at the level of Math 634 (Atiyah-Macdonald), and the first chapter of Hartshorne (especially the first 4 sections).

The course wll start with classical algebraic geometry, as presented in Joe Harris' book: Algebraic Geometry, A first course. This book contains the fundamental constructions and also many beautiful examples.

After we have studied these techniques and examples, we will introduce sheaves and schemes, and revisit these constructions using the language of modern algebraic geometry.

The exact set of topics will depend on the backgrounds and interests of the students. A tentative list of topics includes:

1. Basic constructions in projective geometry, e.g. Segre and Veronese varieties, blowups, rational and birational maps, secant loci, linear joins, quadrics, etc.
2. Grassmannians and Flag varieties.
3. Parameter spaces
4. Dimension and degree of algebraic varieties, including the use of Hilbert functions in algebraic geometry.
5. Smoothness and Zariski tangent spaces
6. Divisors and linear systems
7. Bertini's theorem
8. Sheaves and linear systems
9. Cohomology of sheaves, and the Riemann-Roch formula

There will be biweekly homework sets, and students will present solutions in class.