Complete intersections in simplicial toric varieties

TL;DR

This paper introduces an efficient algorithm to determine whether a simplicial toric ideal is a complete intersection, avoiding explicit generator computation, and classifies certain complete intersection toric varieties.

Contribution

It presents a novel algorithm for checking complete intersections in simplicial toric ideals without computing generators and classifies specific toric varieties over algebraically closed fields.

Findings

Algorithm for complete intersection check in simplicial toric ideals

Simplified algorithm for homogeneous cases

Classification of certain complete intersection toric varieties

Abstract

Given a set $\mathcal A = \{a_1,\ldots,a_n\} \subset \mathbb{N}^m$ of nonzero vectors defining a simplicial toric ideal $I_{\mathcal A} \subset k[x_1,...,x_n]$, where $k$ is an arbitrary field, we provide an algorithm for checking whether $I_{\mathcal A}$ is a complete intersection. This algorithm does not require the explicit computation of a minimal set of generators of $I_{\mathcal A}$. The algorithm is based on the application of some new results concerning toric ideals to the simplicial case. For homogenous simplicial toric ideals, we provide a simpler version of this algorithm. Moreover, when $k$ is an algebraically closed field, we list all ideal-theoretic complete intersection simplicial projective toric varieties that are either smooth or have one singular point.