Centrally symmetric generators in toric Fano varieties
type: Item
instance of: scholarly article
title: Centrally symmetric generators in toric Fano varieties
zbMATH Open document ID: 1039.14024
author: C. Casagrande
published in: Manuscripta Mathematica
publication date: 2003-09-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0210348
review text: This paper tries to reduce the classification of toric Fano varieties to that of lower dimensional ones (as base varieties) by describing clearly the base and the fibre. To be precise, let \(X\) be a smooth, complete toric variety of dimension \(n\), \(N = \Hom_{\mathbb{Z}} (\mathbb{C}^*, T)\) (with \(T\) the maximum torus in \(X\)), \(N_{\mathbb{Q}} = N \otimes_{\mathbb{Z}} \mathbb{Q}\) and \(\Sigma_X\) the fan of \(X\) in \(N_{\mathbb{Q}}\). One defines the set \(G(\Sigma_X)\) of generators of \(\Sigma_X\) as \(\{x \in N \mid x \neq 0, \langle x \rangle \in \Sigma_X\) and \(x\) is primitive in \(\langle x \rangle \cap N\}\). The author proves: Let \(X\) be a smooth toric Fano variety. Suppose that \(G(\Sigma_X)\) contains \(r > 0\) pairs of centrally symmetric generators \(\{x, -x\}\), and let \(H\) be the linear subspace of \(N_{\mathbb{Q}}\) spanned by all such pairs. (1) If \(\dim H < r\), then \(X\) is a toric bundle over a lower dimensional toric Fano variety with fibre a product of del Pezzo varieties. (2) If \(\dim H = n\), then \(X\) is a product of del Pezzo varieties, pseudo del Pezzo varieties and a power of the projective line [the last result extends \textit{G. Ewald}'s earlier result: Discrete Comput. Geom. 3, 49--54 (1988; Zbl 0686.14042)]. The author has other similar results for \(\dim H = n-1\), etc. Readers are advised to consult the paper.
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 52B20
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 14J10
zbMATH DE Number: 1982564
zbMATH Keywords: toric variety
zbMATH Keywords: Fano variety
zbMATH Keywords: del Pezzo variety
reviewed by: De-Qi Zhang
MaRDI profile type: Publication
OpenAlex ID: W1992867098
arXiv ID: math/0210348
DOI: 10.1007/S00229-003-0374-4
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description: scientific article