Commutative Algebra: Set-Theoretic Complete Intersections, Local Cohomology, Free Resolutions, and Rees Rings

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This research project concerns commutative algebra with a view towards algebraic geometry. Often, the most effective method to solve a problem is to create a mathematical model. Frequently, such models involve unknown parameters related by several equations that are often impossible to solve exactly. Commutative algebra is the qualitative study of such systems of polynomial equations. Its applications are far-reaching and include diverse fields such as computer science, cryptography, coding theory, robotics, pattern recognition, and theoretical physics. One can study the sets of solutions of these systems either geometrically or algebraically. This project deals with the algebraic approach. One of the goals of the project is to understand the smallest number of equations needed to describe a geometric object like a curve or surface. Another is to construct the system of equations defining a given geometric object. A third goal is the study of the relations among a given set of polynomial equations, the relations among the relations, and so on. The project involves undergraduate students, graduate students, and postdoctoral fellows in the research. This research project has three main themes. The first is to develop criteria for a variety in projective space to be a set-theoretic complete intersection. A fundamental tool to solve this problem is the theory of local cohomology modules. Local cohomology modules encode the algebraic and topological structure of an algebraic variety. As modules over the ring, local cohomology modules are huge (neither finitely generated nor Artinian), hence intractable. However, as modules over the Weil algebra they can be filtered by simple objects and become manageable. Hence an important task is to understand the D-module structure of local cohomology modules. The second theme is the study of local rings using the notion of distance. This notion was introduced in recent work of the investigator and collaborators to understand the integral closure of ideals. The idea is to use distance as a substitute for shifts in homogeneous resolutions and for the Castelnuovo-Mumford regularity of graded modules. The main goal is to prove general results that are inspired by statements in the graded case. The last theme is to study the implicit equations defining the graph and the image of rational maps between projective spaces. This is a classical problem in elimination theory, commutative algebra, and algebraic geometry with applications, for instance, in geometric modeling.