Toric intersections

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 invariant ${\rm Split}{\rm rad}(I_A)$, defined as the smallest integer $r$ for which there exist toric ideals $I{A_1}, \dots, I_{A_r}$ satisfying $I_A = {\rm rad}(I_{A_1} + \cdots + I_{A_r})$ and $I_{A_i} \neq I_A$ for all $1 \leq i \leq r$. We then compute its exact value in several cases, including the case in which $I_A$ is the toric ideal of a complete bipartite graph. Additionally, we show that ${\rm Split}_{\rm rad}(I_A)$ is equal to the binomial arithmetical rank of $I_A$ when the height of $I_A$ is equal to 2.