Abstract
We study the connection between stringy Betti numbers of Gorenstein toric varieties and the generating functions of the Ehrhart polynomials of certain polyhedral regions. We use this point of view to give counterexamples to Hibi's conjecture on the unimodality of δ-vectors of reflexive polytopes.
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Abramovich, D., Graber, T., Vistoli, A.: Algebraic orbifold quantum products, in Orbifolds in mathematics and physics (Madison, WI, 2001), 1–24, Contemp. Math. 310, Amer. Math. Soc., Providence, RI, 2002
Athanasiadis, C. A.: Ehrhart polynomials, simplicial polytopes, magic squares and a conjecture of Stanley, J. Reine Angew. Math. 583, 163–174 (2005)
Barvinok, A.: A course in convexity, Graduate Studies in Math. 54, Amer. Math. Soc., Providence, RI, 2002
Batyrev, V. V.: Stringy Hodge numbers of varieties with Gorenstein canonical singularities, in Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), 1–32, World Sci. Publishing, River Edge, NJ, 1998
Batyrev, V. V., Dais, D. I.: Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry, Topology 35, 901–929 (1996)
Borisov, L. A., Chen, L., Smith, G. G.: The orbifold Chow ring of toric Deligne-Mumford stacks, J. Amer. Math. Soc. 18, 193–215 (2005)
Chen, W., Ruan, Y.: A new cohomology theory of orbifold, Comm. Math. Phys. 248, 1–31 (2004)
Fernandez, J.: Hodge structures for orbifold cohomology, math.AG/0311026
Fulton, W.: Introduction to toric varieties, Annals of Mathematics Studies, 131, The William H. Roever Lectures in Geometry, Princeton University Press, Princeton, NJ, 1993
Hibi, T.: Ehrhart polynomials of convex polytopes, h-vectors of simplicial complexes, and nonsingular projective toric varieties, Discrete and computational geometry (New Brunswick, NJ, 1989/1990), 165–177, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 6, Amer. Math. Soc. Providence, RI, 1991
Hibi, T.: Algebraic combinatorics on convex polytopes, Carslaw Publications, 1992
Hibi, T.: Star-shaped complexes and Ehrhart polynomials, Proc. Amer. Math. Soc. 123, 723–726 (1995)
Ohsugi, H., Hibi, T.: Special simplicies and Gorenstein toric rings, math.CO/0503666
Poddar, M.: Orbifold cohomology group of toric varieties, in Orbifolds in mathematics and physics (Madison, WI, 2001), 223–231, Contemp. Math. 310, Amer. Math. Soc., Providence, RI, 2002
Stanley, R.P.: Combinatorics and commutative algebra, Progress in Mathematics 41, Birkhäuser, Boston, first edition 1983; second edition 1996
Yasuda, T.: Twisted jets, motivic measures and orbifold cohomology, Compos. Math. 140, 396–422 (2004)
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The first author was partially supported by NSF grant DMS 0500127 and the second author was supported by a Graduate Research Fellowship from the NSF
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Mustaţa, M., Payne, S. Ehrhart polynomials and stringy Betti numbers. Math. Ann. 333, 787–795 (2005). https://doi.org/10.1007/s00208-005-0691-x
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DOI: https://doi.org/10.1007/s00208-005-0691-x