Matroid configurations and symbolic powers of their ideals

TL;DR

This paper introduces matroid configurations, a broad generalization of star configurations, and explores their algebraic properties, including Cohen-Macaulayness of symbolic powers, with applications to hypergraphs, tetrahedral curves, and secant varieties.

Contribution

It develops a new framework for matroid configurations, describes their Hilbert functions and generators, and determines resurgence for various configurations, extending prior knowledge beyond monomial ideals.

Findings

Symbolic powers of matroid configurations are Cohen-Macaulay.

Resurgence is explicitly determined for star and hypersurface configurations.

Connections established between matroid configurations and secant varieties.

Abstract

Star configurations are certain unions of linear subspaces of projective space that have been studied extensively. We develop a framework for studying a substantial generalization, which we call matroid configurations, whose ideals generalize Stanley-Reisner ideals of matroids. Such a matroid configuration is a union of complete intersections of a fixed codimension. Relating these to the Stanley-Reisner ideals of matroids and using methods of Liaison Theory allows us, in particular, to describe the Hilbert function and minimal generators of the ideal of, what we call, a hypersurface configuration. We also establish that the symbolic powers of the ideal of any matroid configuration are Cohen-Macaulay. As applications, we study ideals coming from certain complete hypergraphs and ideals derived from tetrahedral curves. We also consider Waldschmidt constants and resurgences. In particular, we determine the resurgence of any star configuration and many hypersurface configurations. Previously, the only non-trivial cases for which the resurgence was known were certain monomial ideals and ideals of finite sets of points. Finally, we point out a connection to secant varieties of varieties of reducible forms.