Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- \(\mathfrak {X}(G)\) :
-
character lattice of a connected algebraic group \(G\)
- \(Q^*\) :
-
dual polytope to a polytope \(Q\) in a vector space \(V\), i. e. \(Q^* = \{ v \in V^* : \langle u, v \rangle \ge -1\) for every \(u \in Q\}\)
- \(\widehat{F}\) :
-
dual face to a face \(F\) of a polytope \(Q\), i. e. \(\widehat{F} := \{ v \in Q^* : \langle u, v \rangle = -1\) for every \(u \in F\}\)
- \(\mathrm {int}(A)\) :
-
topological interior of a subset \(A\) in some finite-dimensional vector space
- \(\mathrm {relint}(A)\) :
-
relative interior of a subset \(A\) in some finite-dimensional vector space, i. e. topological interior of \(A\) in the affine span of \(A\)
References
Alexeev, V., Brion, M.: Boundedness of spherical Fano varieties, The Fano Conference, pp. 69–80. Univ. Torino, Turin (2004)
Batyrev, V.V.: Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Algebraic Geom. 3(3), 493–535 (1994)
Brion, M.: Groupe de Picard et nombres caractéristiques des variétés sphériques. Duke Math. J. 58(2), 397–424 (1989)
Brion, M.: Vers une généralisation des espaces symétriques. J. Algebra 134(1), 115–143 (1990)
Brion, M.: Variétés sphériques et théorie de Mori. Duke Math. J. 72(2), 369–404 (1993)
Brion, M.: Curves and Divisors in Spherical Varieties, Algebraic Groups and Lie Groups, pp. 21–34. Cambridge Univ. Press, Cambridge (1997)
Brion, M.: The total coordinate ring of a wonderful variety. J. Algebra 313(1), 61–99 (2007)
Casagrande, C.: The number of vertices of a Fano polytope. Ann. Inst. Fourier (Grenoble) 56(1), 121–130 (2006)
Debarre, O.: Fano varieties, Higher Dimensional Varieties and Rational Points (Budapest, 2001). Springer, Berlin (2003)
Demazure, M.: Une démonstration algébrique d’un théorème de Bott. Invent. Math. 5, 349–356 (1968)
Demazure, M.: A very simple proof of Bott’s theorem. Invent. Math. 33(3), 271–272 (1976)
Ewald, G.: Combinatorial Convexity and Algebraic Geometry, Graduate Texts in Mathematics, vol. 168. Springer, New York (1996)
Foschi, A.: Variétés magnifiques et polytopes moment, Ph.D. thesis, Université de Grenoble I (1998)
Fulton, W.: Introduction to Toric Varieties, Annals of Mathematics Studies. Princeton University Press, Princeton (1993)
Gagliardi, G., Hofscheier, J.: Homogeneous spherical data of orbits in spherical embeddings. Transform. Groups. (2015). doi:10.1007/s00031-014-9297-2
Grünbaum, B.: Graduate texts in mathematics. In: Kaibel, V., Klee, V., Ziegler, G.M. (eds.) Convex Polytopes, 2nd edn. Springer, New York (2003)
Hiriart-Urruty, J.-B., Lemaréchal, C.: Fundamentals of Convex Analysis, Grundlehren Text Editions. Springer, Berlin (2001)
Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics. Springer, New York (1978). Second printing, revised
Knop, F., Kraft, H., Luna, D., Vust, T.: Local properties of algebraic group actions, Algebraische Transformationsgruppen und Invariantentheorie. DMV Sem. 13, 63–75 (1989)
Knop, F., Kraft, H., Vust, T.: The Picard group of a \(G\)-variety, Algebraische Transformationsgruppen und Invariantentheorie. DMV Sem. 13, 77–87 (1989)
Knop, F.: The Luna–Vust theory of spherical embeddings, Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989) (Madras), Manoj Prakashan, pp. 225–249 (1991)
Knop, F.: On the set of orbits for a Borel subgroup. Comment. Math. Helv. 70(2), 285–309 (1995)
Knop, F.: Localization of spherical varieties. Algebra Number Theory 8(3), 703–728 (2014)
Luna, D.: Grosses cellules pour les variétés sphériques, Algebraic groups and Lie groups. Austral. Math. Soc. Lect. Ser. 9, 267–280 (1997)
Luna, D.: Variétés sphériques de type \(A\). Publ. Math. Inst. Hautes Études Sci. 94, 161–226 (2001)
Luna, D., Vust, T.: Plongements d’espaces homogènes. Comment. Math. Helv. 58(2), 186–245 (1983)
Mumford, D., Fogarty, J., Clare Kirwan, F.: Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 3rd edn. Springer, Berlin (1994)
Pasquier, B.: Variétés horosphériques de Fano. Bull. Soc. Math. France 136(2), 195–225 (2008)
Rosenlicht, M.: Questions of rationality for solvable algebraic groups over nonperfect fields. Ann. Mat. Pura Appl. (4) 61, 97–120 (1963)
Springer, T.A.: Linear Algebraic Groups, Modern Birkhäuser Classics, 2nd edn. Birkhäuser Boston Inc., Boston (2009)
Timashev, D.A.: Homogeneous spaces and equivariant embeddings, Encyclopaedia of Mathematical Sciences, Invariant Theory and Algebraic Transformation Groups 8. Springer, Heidelberg (2011)
Wasserman, B.: Wonderful varieties of rank two. Transform. Groups 1(4), 375–403 (1996)
Acknowledgments
We would like to thank our teacher Victor Batyrev for encouragement and highly useful advice, as well as Jürgen Hausen for several useful discussions. We are also grateful to Dmitry Timashev for elaborating on Sect. 30.4 of his book. Finally, we thank the referee for several helpful remarks and comments.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gagliardi, G., Hofscheier, J. Gorenstein spherical Fano varieties. Geom Dedicata 178, 111–133 (2015). https://doi.org/10.1007/s10711-015-0047-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-015-0047-y