Multiplicity theory and related topics in commutative algebra

Purdue University, West Lafayette IN

Investigators

Abstract

Linkage, or liaison, has been used as a tool for classifying ideals and projective varieties. Finding necessary and sufficient conditions for two ideals to belong to the same linkage class is the central, though wide open, problem in the theory. The investigator suggests an answer to this question for zero-dimensional monomial ideals. The local-algebraic and the projective-geometric branch of linkage theory have proceeded in parallel, with only few implications known between them. The proposer plans to investigate this subtle interplay by studying the differences between local linkage and homogeneous linkage. Since the equivalence relation generated by classical linkage might be too restrictive for some purposes, the more inclusive notion of Gorenstein linkage has become a welcome alternative. The investigator has the broader goal of showing that in a given regular local ring all Cohen-Macaulay ideals of a fixed codimension belong to the same Gorenstein linkage class. The proposer plans to exploit recent advances on the topic of j-multiplicities to address some long standing problems in equisingularity theory. Equisingularity theory strives to find numerical conditions for when the members of a family of analytic sets are `equivalent' to one another. The investigator seeks to provide such a criterion by proving a general `principle of specialization of integral dependence' based on j-multiplicities. The investigator also proposes an intersection theoretic expression for the j-multiplicity of a module that would yield a recursive formula for computing this multiplicity. The core of an ideal is a subideal that encodes information about all possible reductions of the ideal, while being closely related to Briancon-Skoda type theorems and a conjecture of Kawamata about sections of line bundles. The investigator wishes to explore the connection between cores and multiplier ideals, a notion from complex algebraic geometry. He expects that a better understanding of this interplay will shed new light on both concepts and may lead to a combinatorial description of the core in the monomial case. Having worked out an algorithm for computing the core of zero-dimensional monomial ideals, the investigator now wishes to find such an algorithm for monomial ideals of arbitrary dimension. The known results about cores of ideals require restrictions on the local numbers of generators of the ideal as well as residual or depth conditions, assumptions that are satisfied by any zero-dimensional ideal. The investigator proposes to advance the theory beyond the class of ideals satisfying these conditions, using the monomial case as a testing ground. The proposer works in Commutative Algebra, an area concerned with the qualitative study of systems of polynomial equations in several variables. Such systems arise in algebraic geometry, but also in numerous applications outside of mathematics. Over the past two decades commutative algebraists have become increasingly interested in computational aspects, thereby establishing connections to applied areas such as computer algebra, robotics, cryptography, coding theory, statistics, and biology. This investigator's research too has a strong computational component, and his work on equisingularity theory in particular encompasses concrete geometric applications.