Complete intersections in simplicial toric varieties

Abstract

Given a set $\mathcal{A}=\{a_1,\dots ,a_n\}\subset {\mathbb{N}}^m$ 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.

28 pages, 2 tables. To appear in Journal of Symbolic Computation

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