A note on affine toric varieties
type: Item
instance of: scholarly article
title: A note on affine toric varieties
zbMATH Open document ID: 0986.14032
DOI: 10.1016/S0024-3795(00)00166-X
author: Enrique Reyes
author: Rafael H. Villarreal
author: Leticia Zárate
published in: Linear Algebra and its Applications
publication date: 2001-03-11T00:00:00Z
review text: A matrix \(D=(d_{ij})\) with non-negative integer entries defines a mapping of polynomial rings over a field \(k\) by \(x_i\to t_1^{d_{1i}}, \dots, t_m^{d_{mi}}\). The kernel of this mapping is the toric ideal associated with \(D\). The same matrix also defines a mapping \(Z^n\to Z^m\) whose kernel can be easily determined using the Smith normal form. The main result states that the toric set defined by \(D\) is identical to the variety defined by its toric ideal if and only if two conditions are satisfied, one of which can be formulated in terms of the elementary divisors of \(D\). This is then used (a) to obtain further results for algebraically closed \(k\), (b) to prove that monomial curves over arbitrary fields are affine toric varieties.
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 14R10
zbMATH DE Number: 1547895
zbMATH Keywords: toric set
zbMATH Keywords: toric ideal
zbMATH Keywords: monomial curves
reviewed by: Heinrich Guggenheimer
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1547895