The toroidal embedding arising from an irrational fan
type: Item
instance of: scholarly article
title: The toroidal embedding arising from an irrational fan
zbMATH Open document ID: 0965.14026
DOI: 10.1007/BF03322022
published in: Results in Mathematics
publication date: 2001-07-11T00:00:00Z
review text: Using the language of valuations, Ford generalizes the construction of toric varieties to arbitrary (not necessarily rational) fans \(\Delta\). As in the classical case, the ringed space \(T_N\text{emb}(\Delta)\) comes with an open covering \(\{U_\sigma=T_N\text{emb}(\sigma)\}_{\sigma\in\Delta}\), and the usual semigroup rings \(S_\sigma\) appear as the \(U_\sigma\)-sections of the structure sheaf. However, the big difference is that, in the non-rational case, the \(U_\sigma\) are no longer affine. The set \(\text{Spec} S_\sigma\) is only part of \(U_\sigma\). On the other hand, \(T_N\text{emb}(\Delta)\) keeps most of the nice features we already know from the classical case. The construction is functorial, the space comes with a torus action, and the orbit space equals \(\Delta\) with the usual topology. Moreover, the open subsets \(U_\sigma\) behave well under intersections and the orbit decomposition. It is possible to establish a divisor theory dealing with Weil as well as with Cartier divisors. Finally, Ford presents a discussion of Cox's homogeneous coordinate ring construction. In section 4, there is a second construction of a ringed space out of a fan. Without using the lattice structure \(N\subseteq N_\mathbb R\), one obtains a space \(X(\Delta)\) just by taking the semigroup rings \(k[\sigma^\vee]\) instead of the usual \(S_\sigma=k[\sigma^\vee\cap M]\). In particular, there is always a map \(X(\Delta)\to T_N\text{emb}(\Delta)\) which is studied in the last part of the paper.
reviewed by: Klaus Altmann
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 20M25
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 1294482
zbMATH Keywords: toroidal varieties
zbMATH Keywords: irrational fans
zbMATH Keywords: divisors
zbMATH Keywords: homogeneous coordinate ring
author: Timothy J. Ford
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1294482