Combinatorial symbolic powers

TL;DR

This paper explores the structure of symbolic powers of monomial ideals within combinatorics, highlighting the role of perfect graphs and employing Gr"obner degenerations to unify approaches to determinantal and Pfaffian ideals.

Contribution

It introduces a new framework connecting symbolic powers, perfect graphs, and Gr"obner degenerations, unifying the study of determinantal and Pfaffian ideals.

Findings

Generators of symbolic powers relate to vertex covers in associated graphs.

Perfect graphs are central to understanding symbolic powers in the combinatorial setting.

Gr"obner degenerations reduce complex questions to monomial ideal cases.

Abstract

Symbolic powers are studied in the combinatorial context of monomial ideals. When the ideals are generated by quadratic squarefree monomials, the generators of the symbolic powers are obstructions to vertex covering in the associated graph and its blowups. As a result, perfect graphs play an important role in the theory, dual to the role played by perfect graphs in the theory of secants of monomial ideals. We use Gr\"obner degenerations as a tool to reduce questions about symbolic powers of arbitrary ideals to the monomial case. Among the applications are a new, unified approach to the Gr\"obner bases of symbolic powers of determinantal and Pfaffian ideals.