Toric ideals which are determinantal

TL;DR

This paper introduces a matrix-based approach to characterize the fiber cone of certain monomial ideals, specifically sortable and Freiman ideals, when their defining ideal is generated by quadratic binomials.

Contribution

It provides a new matrix construction that generates the defining ideal of the fiber cone for specific classes of monomial ideals.

Findings

Matrix construction generates the fiber cone ideal.

Characterization of fiber cones for sortable ideals.

Characterization of fiber cones for Freiman ideals.

Abstract

Given any equigenerated monomial ideal $I$ with the property that the defining ideal $J$ of the fiber cone $ F(I)$ of $I$ is generated by quadratic binomials, we introduce a matrix such that the set of its binomial $2$-minors is a generating set of $J$. In this way, we characterize the fiber cone of sortable and Freiman ideals.