MA 764 - Liaison theory and applications (Spring 2005)

Course by Uwe Nagel at the University of Kentucky.

Overview

Schedule

MWF, 9:00 - 9:50, BE 314

Material

A classification theory aims to understand a variety of objects by sorting the objects into disjoint classes such that the elements of a single class all share similar properties. Liaison theory is such a classification theory. It started with the idea of studying a curve by embedding it into a well understood curve such that there is a residual curve. The hope was to do that is such a way that the residual curve is simple enough, so that one can draw conclusions about the original curve. This process of producing a new curve from a given one is called linking. Repeating it leads to an equivalence class that liaison theory tries to understand. Focusing on the defining equations, liaison theory can also be viewed as a classification theory of ideals.

Liaison theory is very well understood in certain cases. There are many efforts to extend these results. This is partially motivated by the numerous applications liaison theory has found. The goal of the course is to present the most important results and open questions as well as to discuss various applications ranging from Combinatorics to Cryptography. The course will be as self-contained as possible though familiarity with the ring- and module-theoretic concepts of a first year graduate Algebra course is preferable. The basic reference for the course will be Migliore's book. In addition, we will use journal articles and recent preprints. Copies of them will be provided.

The grade will be based on active participation in the course and the in-class presentation.

CONTENTS:

2. Saturated ideals and schemes

3. Associated prime ideals and primary decomposition

4. Geometric and algebraic CI-linkage

5. Minimal free resolutions

6. Hom, Ext, and local cohomology

7. Hilbert functions under liaison

8. An application in coding theory

9. Rao's correspondence

10. The Lazarsfeld-Rao property

11. Gorenstein liaison

Literature

J. Migliore, Introduction of Liaison Theory and Deficiency Modules, Birkhäuser, 1998.

J. Migliore, U. Nagel, Liaison and Related Topics: Notes from the Torino Workshop/School, Rend. Sem. Mat. Univ. Politec. Torino 59 (2003), 59--126

W. Bruns, J. Herzog, Cohen-Macaulay rings.

D. Eisenbud, Commutative Algebra with a view towards Algebraic Geometry.

In-class presentations

Sonja Petrovic: Koszul algebras and quadratic Gröbner bases. (

pdf file

Daniel Pinzon: Tropical geometry.

Rachelle Bouchat: Minimal curves. (pdf file)

Keith Kohrs: Generic initial ideals.

Eric Stokes: On the resolution of n+1 general forms. (pdf file)

Philip Busse: Evaluation codes.

David Watson: On the genus of space curves.

Julia Chifman: Toric ideals of phylogenetic invariants.

Dibyajyoti Deb: A bound on the geometric genus.

Last Up-date: May 9, 2005