233a: Theory of Schemes, Fall 2012
and Thursdays 10-11:30, 101b Science Center
Melody Chan, office 242a Science Center
hours: Wednesdays, Fridays 9-10 and by appointment
Assistant: Bao Le Hung, office 421d Science Center
and Harris, Geometry of Schemes
Foundations of Algebraic Geometry (available online)
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
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.
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
Problem Set 2, due
Problem Set 3, due
Problem Set 4, due
Problem Set 5, due
Problem Set 6, due
Problem Set 7, due
Problem Set 8, due
Problem Set 9, due
Problem Set 10, due
Problem Set 11, due
Problem Set 12, due
I urge you to collaborate on
problem sets. Please write up your solutions separately and indicate
with whom you collaborated.
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.
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.
There are many other
classes/seminars that might interest you. For example:
Baby Algebraic Geometry seminar,
Algebraic Geometry seminar, Tuesdays at 3pm
Math 266y, Geometry of families
of curves, Joe Harris, Fall 2012, MWF at 10am
Math 285y, Tropical geometry,
Melody Chan, Spring 2013
conference, Oct 26-28 http://www.agneshome.org/brown-2012