Gorenstein spherical Fano varieties. (English) Zbl 1349.14165
Summary: We obtain a combinatorial description of Gorenstein spherical Fano varieties in terms of certain polytopes, generalizing the combinatorial description of Gorenstein toric Fano varieties by reflexive polytopes and its extension to Gorenstein horospherical Fano varieties due to Pasquier. Using this description, we show that the rank of the Picard group of an arbitrary \(d\)-dimensional \(\mathbb Q\)-factorial Gorenstein spherical Fano variety is bounded by \(2d\). This paper also contains an overview of the description of the natural representative of the anticanonical divisor class of a spherical variety due to Brion.
MSC:
| 14M27 | Compactifications; symmetric and spherical varieties |
| 14J45 | Fano varieties |
| 14L30 | Group actions on varieties or schemes (quotients) |
| 52B20 | Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) |
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