Integrality, Blowup Algebras and Multiplicities

Purdue University, West Lafayette IN

Investigators

Abstract

Passing from an affine algebra or an ideal to the integral closure is a fundamental operation. The investigator intends to examine computational aspects of this process. While there exist algorithms for computing the integral closure of an affine domain, little is known about the complexity of such calculations. To address this problem the investigator intends to find estimates for the number of algebra generators of the integral closure and for the degrees of these generators. In the case of ideals on the other hand, one still has no efficient method for computing integral closures. In view of this lack of an algorithm, the investigator intends to express, or at least approximate, the integral closure of an ideal by a sum of colon ideals that are more readily accessible. The core of an ideal is a counterpart of the integral closure: it encodes information about all possible reductions of a given ideal, i.e., all ideals over which the given ideal is integral. The investigator intends to work on a conjectural formula that, if proved, would give an explicit expression for the core of a large class of ideals. Continuing the somewhat computational theme the investigator proposes to establish bounds for the Castelnuovo-Mumford regularity of embedded projective varieties in terms of their defining equations. The known results require strong assumptions on the singularities of the variety which the investigator hopes to weaken. In this vein he intends to show that rational singularities persist under generic linkage of ideals in the linkage class of a complete intersection. In continuation of his earlier work on blowup algebras the investigator plans to study the rings representing special fibers of blowups. These `special fiber rings' describe, for instance, images of rational maps, including secant varieties and Gauss images. The investigator intends to study the Cohen-Macaulayness of special fiber rings, estimate their depth and compute multiplicities. Suitable depth estimates would have a bearing on a conjecture by Mazur that originates in Wiles' work on deformations of Galois representations. Formulas for the multiplicity on the other hand would lead to an extension of the Teissier-Pluecker formula for the degrees of certain dual varieties. The investigator works in the area of Mathematics called "Commutative Algebra," which deals with the qualitative study of systems of polynomial equations in several variables. Such systems arise in numerous applications outside mathematics. Over the past two decades commutative algebraists have become more concerned with computational aspects, thereby emphasizing connections to applied areas such as computer algebra, robotics, cryptography and coding theory. This project addresses both theoretical and computational aspects of commutative algebra.