Varieties with torus actions: algebra and combinatorics

University Of Connecticut, Storrs CT

Investigators

Abstract

This project deals with studying the defining equations and higher syzygies of toric varieties, i.e., equivariant compactifications of algebraic tori. In general, it is difficult to find explicit equations for a given embedding. However, often one can determine the degrees of the defining equations as well as the degrees of the higher syzygies, and this information reveals geometric properties of the embedding. For example, the existence of nonlinear minimal syzygies is related to the existence of secant planes. An embedding of a toric variety into projective space corresponds to a lattice polytope, and the goal of this project is to further the understanding of the interplay between the defining equations and syzygies of the embedding and the combinatorial properties of the polytope. Explicit results for toric varieties can also be useful in giving bounds of the degrees of the generators and higher syzygies of varieties that are not toric, but admit a degeneration to a toric variety. For example, we apply this to the moduli space of points on the projective line. Algebraic geometry is an area of mathematics that interacts with many other areas, such as number theory, topology, mathematical physics, and combinatorics. The most basic building blocks of the spaces studied in algebraic geometry are vanishing sets of polynomial equations. However, in general, the space in question is given abstractly, and often one does not know what these polynomial equations are. This project is about studying this question for a special class of geometric spaces that exhibit additional combinatorial structure allowing a more concrete approach than in the general case.