Math 233a: Theory of Schemes, Fall 2012
Tuesdays and Thursdays 10-11:30, 101b Science Center

Instructor: Melody Chan, office 242a Science Center
email: mtchan@math.harvard.edu
office hours: Wednesdays, Fridays 9-10 and by appointment
course website: www.math.harvard.edu/~mtchan/2012Fall_233a.html

Course Assistant: Bao Le Hung, office 421d Science Center
email: lehung@math.harvard.edu
Problem session: TBD

Textbooks:
Eisenbud and Harris, Geometry of Schemes
Hartshorne, Algebraic Geometry
Vakil, Foundations of Algebraic Geometry (available online)

Topics:
This class is the first semester of a year-long introduction to the theory of schemes in algebraic geometry. The second semester is expected to be offered in 2014. We will follow the textbook Geometry of Schemes for the most part. In addition to building the foundations of scheme theory, we will emphasize examples and classical constructions.

Topics will include: affine schemes, projective schemes, morphisms, sheaves of modules, classical constructions, the functor of points, Hilbert schemes, and more as time permits.

Week 1: Presheaves and sheaves, affine schemes.
Week 2: Schemes in general, subschemes, morphisms.
Week 3: Fiber products, examples of multiple points, schemes over nonalgebraically closed fields, local schemes.
Week 4: Primary decomposition, a first look at flatness, examples of arithmetic schemes.
Week 5: The fiber product again, finite, finite type, and separated morphisms.
Week 6: Separated and proper, the Proj construction, closed subschemes of Proj.
Week 7: Projective morphisms are proper, global Spec and Proj.
Week 8: Tangent cones, Hilbert polynomials and flatness.
Week 9: Locally free sheaves, inverse image and pullback sheaves, morphisms to projective space.
Week 10: Weil and Cartier divisors, examples, complete intersections.
Week 11: Bezout's theorem, Cohen-Macaulay schemes, blow ups.
Week 12: Blow ups, Fano schemes. Thanksgiving.
Week 13: Functor of points, representability, Hilbert schemes, multigraded Hilbert schemes.
Week 14: First order deformations, tangent spaces to Grassmannians and Fano schemes.

Homework:
Weekly problem sets constituting 100% of the course grade, due on Thursdays starting September 13.
No homework will be accepted after Tuesday, December 11 at 5pm.
Problem Set 1, due September 13
Problem Set 2, due September 20
Problem Set 3, due September 27
Problem Set 4, due October 4
Problem Set 5, due October 11
Problem Set 6, due October 18
Problem Set 7, due October 25
Problem Set 8, due November 1
Problem Set 9, due November 8
Problem Set 10, due November 15
Problem Set 11, due November 27
Problem Set 12, due December 11

Collaboration:
I urge you to collaborate on problem sets. Please write up your solutions separately and indicate with whom you collaborated.

Prerequisites:
This class is intended to be suitable for those with no prior knowledge of schemes. The official prerequisites are Math 221 (commutative algebra) and 232a (introduction to algebraic geometry.) For my purposes, a prior course in algebraic geometry is not strictly necessary. However, I would like you to have done some commutative algebra and have good working knowledge of things like localization, primary decomposition, and dimension theory. Good references are Atiyah and Macdonald Introduction to Commutative Algebra and Eisenbud Commutative Algebra with a View Toward Algebraic Geometry. If you would like to take this class but have less background, please do talk to me in any case to see if we can work something out.

Accommodations for students with disabilities
If you need accommodations for a disability, please talk to me as soon as possible and within the first two weeks of the term.

Other goings on:
There are many other classes/seminars that might interest you. For example:
Baby Algebraic Geometry seminar, time TBD
Harvard/MIT Algebraic Geometry seminar, Tuesdays at 3pm http://math.mit.edu/seminars/ags/
Math 266y, Geometry of families of curves, Joe Harris, Fall 2012, MWF at 10am
Math 285y, Tropical geometry, Melody Chan, Spring 2013
AGNES conference, Oct 26-28 http://www.agneshome.org/brown-2012