Radical splittings of toric ideals

TL;DR

This paper characterizes when toric varieties can be expressed as intersections of other toric varieties, introduces the radical splitting number, and computes it for various cases, especially graph-based toric ideals.

Contribution

It provides a necessary and sufficient condition for set-theoretic intersections of toric varieties and introduces the radical splitting number with explicit calculations for graph-related ideals.

Findings

Radical splitting number equals 3 for complete bipartite graph toric ideals.

Radical splitting number matches the binomial arithmetical rank when height is 2.

Characterization of toric varieties as intersections of other toric varieties.

Abstract

Let $I_A \subset K[x_1,\ldots,x_n]$ be a toric ideal. In this paper, we provide a necessary and sufficient condition for the toric variety $V(I_A)$, over an algebraically closed field, to be expressed as the set-theoretic intersection of other toric varieties. We also introduce the radical splitting number of $I_A$, denoted by ${\rm Split}_{\rm rad}(I_A)$, and compute its exact value in several cases, with particular emphasis on toric ideals arising from graphs. In particular, we show that ${\rm Split}_{\rm rad}(I_A)=3$ for toric ideals of complete bipartite graphs. Additionally, we prove that ${\rm Split}_{\rm rad}(I_A)$ coincides with the binomial arithmetical rank of $I_A$ when the height of $I_A$ is equal to 2.