Rees algebras and singularities

Purdue University, West Lafayette IN

Investigators

Abstract

A general goal in equisingularity theory is to provide criteria for a family of analytic sets to be topologically trivial. Ideally, such criteria involve numerical data, like multiplicities, that only depend on the individual members rather than the total space of the family. In prior work the investigator established a sufficient condition for the Whitney equisingularity of families of arbitrary isolated singularities, using the new notion of epsilon multiplicity as numerical invariant. Now he wishes to prove the necessity of his condition for equisingularity, which would result in a fiber-wise numerical characterization of Whitney equisingularity in the case of isolated singularities. In addition, he intends to advance the general theory of epsilon multiplicity beyond the context of equisingularity theory. The investigator proposes a program to study rational curves in projective space, most notably rational plane curves, through the syzygy matrix of the forms parametrizing them. Solely from the syzygy matrix, he wishes to extract local information about the singularities of the curve and understand the global positioning of these singularities. In particular, he proposes to set up a correspondence between the types of singularities on the one hand and the shapes of the syzygy matrix on the other hand, and to use this correspondence to stratify the space of rational plane curves of a given degree. The investigator plans to continue his work on Rees algebras of ideals by studying the implicit equations of such algebras. Understanding or finding these equations is a fundamental and difficult problem in elimination theory that is wide open even for the simplest of ideals. The investigator intends to focus on ideals whose generators parametrize projective varieties. In order to bound the degrees of the implicit equations and to understand the Castelnuovo-Mumford regularity of the Rees algebra, he wishes to prove that the Rees algebra and the homogeneous coordinate ring of the variety have the same regularity. The investigator has the long-term goal to determine the defining equations explicitly if the parametrized variety is a rational plane curve. The proposed research is in the area of Commutative Algebra, a field of mathematics that has its roots in the qualitative study of systems of polynomial equations in several variables. Commutative Algebra has close ties to geometry, a connection that is prominent in the investigator's projects on equisingularity and rational curves. Systems of polynomial equations also arise in numerous applications outside of mathematics. The investigator's project on implicit equations of Rees algebras encompasses this applied aspect. In particular, the problem of finding implicit equations of surfaces defined parametrically has relevance in geometric modeling and computer-aided design, where it is known as implicitization problem.