Strongly symmetric smooth toric varieties
In the paper under review, the authors classify the complete smooth toric varieties defined by hyperplane arrangements, and show that they are indeed projective.NEWLINENEWLINELet \(N\) be a free \(\mathbb Z\)-module of rank \(r\) and let \(\Sigma\) be a complete fan in \(V = N \otimes \mathbb R\). The fan \(\Sigma\) is called \textit{strongly symmetric} if the union of its codimension one cones decomposes in the union of some hyperplanes \(H_1, \ldots, H_n\) of \(V\). The complete toric variety \(X_\Sigma\) associated to \(\Sigma\) is then called a \textit{strongly symmetric toric variety}, and we may think it as the toric variety attached to the hyperplane arrangement \(\mathcal A(\Sigma) = \{H_1, \ldots H_n\}\).NEWLINENEWLINELet \(\Sigma\) be a strongly symmetric fan, set \(\mathcal A = \mathcal A (\Sigma)\) and denote \(\mathcal K(\mathcal A)\) the set of \textit{chambers} of \(\mathcal A\), namely the connected components of \(V \smallsetminus \bigcup_{H \in \mathcal A} H\). Given \(H_i \in \mathcal A\), choose \(\alpha_i \in V^*\) such that \(H_i = \alpha_i^\perp\) and set \(R =\{\pm\alpha_1, \ldots, \pm\alpha_n\}\). If \(K \in \mathcal K(\mathcal A)\), let \(B^K \subset R\) be the subset of the elements which vanish on a codimension one face of \(K\) and which are positive on \(K\). Then \(X_\Sigma\) is smooth if and only if the couple \((\mathcal A, R)\) is a \textit{crystallographic arrangement}, namely for every chamber \(K \in \mathcal K(\mathcal A)\) it holds the inclusion \(R \subset \sum_{\alpha \in B^K} \mathbb Z \alpha\).NEWLINENEWLINEExamples of crystallographic arrangements are those defined by the root systems of classical semisimple Lie algebras, in which case \(\mathcal K(\mathcal A)\) coincides with the set of Weyl chambers. A certain generalization of the universal enveloping algebras of Lie algebras yields Hopf algebras to which one can associate root systems and Weyl groupoids, and it turns out that finite Weyl groupoids correspond to crystallographic arrangements. Therefore the classification of smooth strongly symmetric toric varieties follows by the classification of finite Weyl groupoids given by the first author together with \textit{I.~Heckenberger} [J. Reine Angew. Math. 702, 77--108 (2015; Zbl 1358.20037)].
scientific article; zbMATH DE number 6081385
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