2025/07/21 by Katsabekis, Anargyros, Thoma, Apostolos
#05C25 #05E40 #13F65 #14M25 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2507.15785
Let IA ⊂ K[x1,…,xn] be a toric ideal. In this paper, we provide a necessary and sufficient condition for the toric variety V(IA), 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(IA), defined as the smallest integer r for which there exist toric ideals IA1, …, IAr satisfying IA = \rm rad(IA1 + ⋯ + IAr) and IAi ≠ IA for all 1 ≤ i ≤ r. We then compute its exact value in several cases, including the case in which IA is the toric ideal of a complete bipartite graph. Additionally, we show that \rm Split\rm rad(IA) is equal to the binomial arithmetical rank of IA when the height of IA is equal to 2.