Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

The rational homology of toric varieties is not a combinatorial invariant
HTML articles powered by AMS MathViewer

by Mark McConnell PDF
Proc. Amer. Math. Soc. 105 (1989), 986-991 Request permission

Abstract:

We prove that the rational homology Betti numbers of a toric variety with singularities are not necessarily determined by the combinatorial type of the fan which defines it; that is, the homology is not determined by the partially ordered set formed by the cones in the fan. We apply this result to the study of convex polytopes, giving examples of two combinatorially equivalent polytopes for which the associated toric varieties have different Betti numbers.
References
  • A. Ash, D. Mumford, M. Rapoport, and Y. Tai, Smooth compactification of locally symmetric varieties, Lie Groups: History, Frontiers and Applications, Vol. IV, Math Sci Press, Brookline, Mass., 1975. MR 0457437
  • Louis J. Billera and Carl W. Lee, A proof of the sufficiency of McMullen’s conditions for $f$-vectors of simplicial convex polytopes, J. Combin. Theory Ser. A 31 (1981), no. 3, 237–255. MR 635368, DOI 10.1016/0097-3165(81)90058-3
  • V. I. Danilov, The geometry of toric varieties, Uspekhi Mat. Nauk 33 (1978), no. 2(200), 85–134, 247 (Russian). MR 495499
    • J. Fine, Geometric progressions, convex polytopes, and torus embeddings (preprint).
  • Richard P. Stanley, The number of faces of a simplicial convex polytope, Adv. in Math. 35 (1980), no. 3, 236–238. MR 563925, DOI 10.1016/0001-8708(80)90050-X
    • —, Generalized $H$-vectors, intersection cohomology of toric varieties, and related results, Commutative Algebra and Combinatorics, Adv. Stud. Pure Math., vol. 11, ed. M. Nagata and H. Matsumura, pub. Kinokuniya, Tokyo and New York, (1987), pp. 187-213.
  • G. Kempf, Finn Faye Knudsen, D. Mumford, and B. Saint-Donat, Toroidal embeddings. I, Lecture Notes in Mathematics, Vol. 339, Springer-Verlag, Berlin-New York, 1973. MR 0335518
Similar Articles
Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 105 (1989), 986-991
  • MSC: Primary 14L32; Secondary 14F45, 14J40, 52A25, 52A37
  • DOI: https://doi.org/10.1090/S0002-9939-1989-0954374-9
  • MathSciNet review: 954374