Math 465: Topics in Algebra
Cohomology and computation in algebraic geometry
MWF 9:00
Description:
This course is devoted to the
computational techniques of modern algebraic geometry.
Here is a sample problem:
Given four points in the plane, how many quadric polynomials
Ax2+Bxy
+Cy2+Dx+Ey+F
vanish at all these points? For instance, every quadric
vanishing at (0,0),(1,0),(0,1), and (1,1) is a linear
combination
c1y(y-1) + c2x(x-1).
But what happens when the four points are collinear?
There are three independent quadrics vanishing
at (0,0),(1,0),(-1,0), and (2,0):
c1y(y-1) +
c2y(x+2) +
c3y(x-3).
The dimenison of the space of vanishing quadrics jumps from two to three.
Sheaf cohomology is a tool for systematically analyzing this jumping phenomenon.
While the definitions may seem abstract at first, our approach will focus
on concrete algebraic and geometric problems. In particular,
we will include algorithms for computing cohomology groups in examples.
Here are some specific topics we will explore:
graded rings and modules, generators,
Hilbert syzygy theorem, resolutions, Betti numbers,
algorithmic computations;
general theory of sheaves, coverings,
Cech cocycles and cohomology;
affine varieties, coherent
sheaves, vanishing of cohomology;
projective varieties, twisted sheaves,
correspondence between graded modules and coherent sheaves
on projective space;
Hom and Ext, Serre duality;
sheaves of differential forms, explicit computation of Betti
numbers of complex projective varieties.
Prerequisites:
This course should be accessible to all graduate students
and motivated undergraduates.
It will assume the definitions and
elementary properties of rings, ideals, and polynomials.
A semester course in modern algebra should be
sufficient. A previous course in
topology, algebraic geometry, or complex analysis
might increase the resonance of
the material, but is not necessary to do well in the class.
References: The main reference will be lecture notes
distributed before each lecture. One good reference for
graded modules and resolutions is:
D. Eisenbud,
Commutative algebra:
With a view toward algebraic
geometry,
Springer-Verlag, New York, 1995
I learned coherent sheaves from Chapters 2 and 3 of
R. Hartshorne,
Algebraic geometry,
Springer-Verlag, New York, 1977
This is an essential reference for anyone considering advanced study
of modern algebraic geometry. The most elegant account is:
J.P.Serre, Faisceaux algébriques cohérents,
Annals of Mathematics 61, No. 2 (1955),
197-278
Anyone planning to study mathematics in graduate school should
consider learning French from this beautiful paper.
Undergraduate research opportunities: This course
will be excellent preparation for
VIGRE
summer research opportunities in algebraic geometry.
Contact information:
Brendan Hassett
Office: Herman Brown 422
Phone: (713) 348-5261
Email: hassett@math.rice.edu
Webpage:
http://www.math.rice.edu/~hassett