Radical splittings of toric ideals

TL;DR

This work addresses when a toric variety $V(I_A)$ can be written as a set-theoretic intersection of toric varieties by studying radical splittings of toric ideals. It develops a constructive radical splitting criterion based on a minimal binomial generating set and partitions of $\ker_{\,\mathbb{Z}}(A)$, linking the radical splitting number ${\rm Split}_{\mathrm{rad}}(I_A)$ to the binomial arithmetical rank $bar(I_A)$ and the height ${\rm ht}(I_A)$. The authors compute exact values in key cases, notably ${\rm Split}_{\mathrm{rad}}(I_{K_{m,n}})=3$ for complete bipartite graphs and ${\rm Split}_{\mathrm{rad}}(I_A)={\rm bar}(I_A)$ when ${\rm ht}(I_A)=2$, and they prove equality ${\rm Split}_{\mathrm{rad}}(I_G)={\rm Split}(I_G)$ for bipartite graphs. The results yield practical invariants for toric ideals, with implications for graph theory, combinatorial commutative algebra, and set-theoretic generation up to radical, including constructions where the ordinary splitting number grows without bound while the radical splitting number remains fixed.

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.

Theorems & Definitions (24)

• Definition 1.1
• Remark 1.2
• Theorem 2.1
• Corollary 2.2
• Proposition 2.3
• Remark 2.4
• Theorem 2.5
• Corollary 2.6
• Remark 2.7
• Proposition 2.8