Problems in Commutative Algebra: Free Resolutions, Multiplicities, and Blowup Rings

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This project is in commutative algebra, an area of pure mathematics with close ties to algebraic geometry. Commutative algebra deals, in a wider sense, with the qualitative study of systems of polynomial equations in several unknowns. Solutions of such systems play an important role in many areas of science and engineering, hence the proposed research has applied aspects as well. One of its objectives is to devise numerical criteria for when the solution sets of different systems of equations are "alike," another one aims at constructing systems of equations that have a given geometric object, like a curve or a surface, as its solution set. This second problem is also relevant for applications in geometric modeling and computer-aided design. A third goal of the project is to further develop a fundamental tool in algebra, called "free resolutions." Free resolutions are a method to study complex algebraic structures by means of a possibly infinite sequence of simpler objects, namely, matrices.In more technical terms, the topics of this project are equisingularity theory, the implicitization problem for Rees algebras, algebraic properties of multiplier ideals, the extension to local rings of results and techniques that are inherent to the graded setting. To be more specific, a general goal in equisingularity theory is to devise fiberwise numerical criteria for when a family of analytic spaces is topologically trivial, is Whitney equisingular, or satisfies other equisingularity conditions. With Kleiman and Validashti the PI proved that a family of isolated singularities is Whitney equisingular, and hence topologically trivial, if a newly defined generalized multiplicity, the epsilon multiplicity, is constant across the family. Now the PI plans to examine which stronger equisingularity conditions the constancy of the epsilon multiplicity implies. A related, fundamental problem is to understand singularities whose Jacobian module has epsilon multiplicity zero. Equisingularity conditions are closely related to numerical criteria for integral dependence of modules, and the PI wishes to devise such a criterion based on intersection numbers. A classical problem in elimination theory is to determine the defining ideals of Rees algebras, which gives, in particular, the implicit equations of graphs and images of rational maps between projective spaces. The PI plans to work on this problem for regular maps parametrizing a surface or, more generally, for rational maps with codimension three base locus. In recent joint work with Corso, Huneke, and Polini, the PI introduced the notion of distance in free resolutions over local rings and used it to study integral closures of ideals. Now the PI proposes this notion as a substitute for the shifts in graded free resolutions, in order to extend known results from the graded to the local case -- such as criteria for the Cohen-Macaulay and Gorenstein properties of associated graded rings and bounds for the Loewy length of modules having finite projective dimension. Another of the PI's goals is tostudy algebraic properties of (Mather-Jacobian) multiplier ideals on singular varieties and to determine which ideals can be realized as multiplier ideals.