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Some combinatorics of binomial coefficients and the Bloch-Gieseker property for some homogeneous bundles

Author: Mei-Chu Chang
Journal: Trans. Amer. Math. Soc. 354 (2002), 975-992
MSC (2000): Primary 14F05; Secondary 14J60, 05A10
Published electronically: September 19, 2001
MathSciNet review: 1867368
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Abstract: A vector bundle has the Bloch-Gieseker property if all its Chern classes are numerically positive. In this paper we show that the non-ample bundle $\Omega ^{p}_{\mathbb{P}_{n}}(p+1)$ has the Bloch-Gieseker property, except for two cases, in which the top Chern classes are trivial and the other Chern classes are positive. Our method is to reduce the problem to showing, e.g. the positivity of the coefficient of $t^{k}$ in the rational function $\frac{(1+t)^{\binom n p} (1+3t)^{\binom {n}{p-2}} \cdots (1+(p-1)t)^{\binom n2}... ...1+2t)^{\binom {n}{p-1}} (1+4t)^{\binom {n}{p-3}} \cdots (1+pt)^{\binom {n}{1}}}$ (for $p$ even).

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Additional Information

Mei-Chu Chang
Affiliation: Department of Mathematics, University of California, Riverside, California 92521

Received by editor(s): September 10, 2000
Published electronically: September 19, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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