Commutative Algebra, Algebraic Geometry, Computation and Statistical applications

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

David Eisenbud will perform research on a number of areas in which algebraic geometry mixes with homological commutative algebra. One of these is the study of the Castelnuovo-Mumford regularity of high powers of ideals. For an ideal in the homogeneous coordinate ring of a projective variety that is of finite colength and is generated by forms of a single degree, this "assymptotic regularity" is connected with the properties of fibers of the corresponding map of varieties, and Eisenbud will use this connection to investigate generic projections. A second area is the study of schemes that are 2-regular. This work builds on the classification for reduced schemes that was recently completed by Eisenbud, Green, Hulek and Popescu. A third area has to do with the computation of higher direct images using exterior algebra methods. Eisenbud will study certain varieties that appear naturally in the deformation spaces of bundles in this way. All these areas are supported by computations based on Groebner basis methods. Eisenbud will collaborate with Stillman and Grayson in the development of the Macaulay2 program, which is currently the best tool for computations in projective geometry. Finally, Eisenbud will collaborate on a project with Diaconis, Holmes, and their students involving the application of techniques in algebraic geometry to statistics. An algebraic variety in complex n-space is a set defined by the simultaneous vanishing of a collection of polynomial functions of n variables. Many of the important geometric objects that appear in mathematics can be defined this way. One way to study an algebraic variety is to ask for the dimension of the vector space of polynomials of degree at most d that vanish on the variety, as a function of d. This is called the Hilbert function of the variety. David Hilbert showed how to compute this function, and refine it, by computing something called a free resolution of the variety. My work uses such free resolutions to study varieties in many contexts. For example, if a variety of dimension n is embedded in a vector space of dimension n+1, then the variety is defined by just one equation, a relatively simple case. Most varieties of dimension n can not be embedded in this way, but any variety of dimension n can be mapped linearly onto a variety of the same dimension in an n+1 dimensional space. One of the questions I will study is the difference between the original variety and its image, for the best possible embeddings.