Cohomological and Combinatorial Methods in the Study of Symbolic Powers and Equations defining Varieties

TL;DR

This thesis explores cohomological and combinatorial techniques to analyze symbolic powers and defining equations of algebraic varieties, connecting algebraic geometry, representation theory, and combinatorics.

Contribution

It introduces new methods for studying cohomology vanishing, connectedness during deformations, minors of matrices, and Cohen-Macaulay properties of symbolic powers.

Findings

Vanishing conditions for cohomology modules identified.

Connectedness behavior during Groebner deformations analyzed.

Relations between minors of generic matrices established.

Abstract

In this PhD thesis we will discuss some aspects in Commutative Algebra which have interactions with Algebraic Geometry, Representation Theory and Combinatorics. In particular, in the first chapter we will focus on understanding when certain cohomology modules vanish, a classical problem raised by Grothendieck. In the second chapter we will use local cohomology to study the connectedness behavior during a Groebner deformation and the arithmetical rank of certain varieties. In the third chapter, we will investigate the relations between the minors of a fixed size of a generic matrix by using tools from the representation theory of the general linear group (the results of this chapter will appear in a joint paper with Bruns and Conca). In the last chapter we will use combinatorial methods to study the Cohen-Macaulay property of the symbolic powers of Stanley-Reisner ideals. In the thesis are included five appendixes with some basic needed facts and a preliminary chapter introducing to local cohomology.