MA 765 - Geometry of Syzygies (Spring 2006)
Course by Uwe Nagel
at the University of Kentucky.
Overview
Schedule
MWF, 3:00 - 3:50 pm, FB 213.
Material
This will be a course about recent developments at the intersection of
Commutative Algebra and Geometry. Consider the set X of common zeros of
a set of homogeneous polynomials. Then the (first) syzygies of X are the
relations between the given polynomials. More generally, syzygies are
solutions of linear
system of equations where the coefficient matrix has entries in a
ring. Typically the ring will be a polynomial ring.
The consideration of syzygies has been initiated by Hilbert. The fact
that the extremely algebraic notion of syzygies is so intimately
related to the geometry of X is at the heart of the course.
The focus will be the case where X is a curve.
CONTENTS:
1. Free resolutions and Hilbert functions
2. Multilinear algebra
3. Varieties of minimal degree
4. Finite sets
5. Castelnuovo-Mumford regularity
6. A regularity bound for curves
7. The linear strand of a resolution
- D. Eisenbud, Geometry of syzygies, Graduate Texts in
Mathematics 229, Springer,
2005.
- W. Bruns, J. Herzog, Cohen-Macaulay rings, Cambridge
Studies in Advanced Mathematics 39, Cambridge University
Press, 1998.
- D. Eisenbud, Commutative Algebra, Graduate Texts in
Mathematics 150, Springer,
2005.
Rachelle Bouchat: The New Intersection Theorem
Philip Busse: A bound for the regularity of
fat points in linearly general position
Erik Stokes: Exterior algebra methods for the minimal
free resolution conjecture
Sonja Petrovic: Castelnuovo's lemma
Last Up-date: April 30, 2006