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The rational homology of toric varieties is not a combinatorial invariant
Author:
Mark McConnell
Journal:
Proc. Amer. Math. Soc. 105 (1989), 986-991
MSC:
Primary 14L32; Secondary 14F45, 14J40, 52A25, 52A37
MathSciNet review:
954374
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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.
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P. Stanley, The number of faces of a simplicial convex
polytope, Adv. in Math. 35 (1980), no. 3,
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- L. J. Billera, C. W. Lee, Sufficiency of McMullen's conditions for
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- V. I. Danilov, The geometry of toric varieties, Russian Math. Surveys 33:2 (1978), 97-154. MR 495499 (80g:14001)
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- J. Fine, Geometric progressions, convex polytopes, and torus embeddings (preprint).
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- R. Stanley, The number of faces of a simplicial convex polytope, Adv. in Math. 35 (1980), 236-238. MR 563925 (81f:52014)
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- -, Generalized -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.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9939-1989-0954374-9
PII:
S 0002-9939(1989)0954374-9
Keywords:
Polytopes,
toric varieties,
 -vector
Article copyright:
© Copyright 1989 American Mathematical Society
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