On projective toric varieties whose defining ideals have minimal generators of the highest degree.
type: Item
instance of: scholarly article
title: On projective toric varieties whose defining ideals have minimal generators of the highest degree.
zbMATH Open document ID: 1069.14057
author: Shoetsu Ogata
publication date: 2004-08-20T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116098
review text: Given an \(n\)-dimensional, projectively normal, toric variety \(X\), then it is well-known by Sturmfels that its (binomial) ideal is generated in degree \(\leq(n+1)\). Generalizing \textit{R.J. Koelman}'s papers [Beitr. Algebra Geom. 34, No. 1, 57--62 (1993; Zbl 0781.14025); Tohoku Math. J., II. Ser. 45, No. 3, 385--392 (1993; Zbl 0809.14042)] (dealing with the case \(n=2\)), the author classifies those cases, i.e. lattice polytopes, that require generators of degree \(n+1\). The result is that a lattice polytope enforces generators of degree \(n+1\) if and only if it is a simplex with ``smooth'' facets but containing interior lattice points. However, reading p.~2245 of the paper, I got the impression that the author has used the fact that so-called empty simplices (the vertices are the only lattice points) are smooth (isomorphic to the standard simplex), which is not true.
reviewed by: Klaus Altmann
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 52B20
zbMATH DE Number: 2093471
MaRDI profile type: Publication
OpenAlex ID: W2476961580
cites work: Q3125884
cites work: Binomial ideals
cites work: On the ampleness of invertible sheaves in complete projective toric varieties
cites work: Q4123438
cites work: Introduction to Toric Varieties. (AM-131)
cites work: Q3698928
cites work: Generators for the ideal of a projectively embedded toric surface
cites work: A criterion for the ideal of a projectively embedded toric surface to be generated by quadrics
cites work: Varieties Defined by Quadratic Equations
cites work: On generators of ideals defining projective toric varieties
cites work: Q4039830
cites work: On quadratic generation of ideals defining projective toric varieties
cites work: Q4861423
cites work: Equations Defining Toric Varieties
DOI: 10.5802/AIF.2005
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 2093471