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April, 2015 Differentials of Cox rings: Jaczewski's theorem revisited
Oskar KĘDZIERSKI, Jarosław A. WIŚNIEWSKI
J. Math. Soc. Japan 67(2): 595-608 (April, 2015). DOI: 10.2969/jmsj/06720595

Abstract

A generalized Euler sequence over a complete normal variety $X$ is the unique extension of the trivial bundle $V \otimes {\mathcal O}_X$ by the sheaf of differentials $\Omega_X$, given by the inclusion of a linear space $V\subset {\rm Ext}^1_X({\mathcal O}_X,\Omega_X)$. For $\Lambda$, a lattice of Cartier divisors, let ${\mathcal R}_\Lambda$ denote the corresponding sheaf associated to $V$ spanned by the first Chern classes of divisors in $\Lambda$. We prove that any projective, smooth variety on which the bundle ${\mathcal R}_\Lambda$ splits into a direct sum of line bundles is toric. We describe the bundle ${\mathcal R}_\Lambda$ in terms of the sheaf of differentials on the characteristic space of the Cox ring, provided it is finitely generated. Moreover, we relate the finiteness of the module of sections of ${\mathcal R}_\Lambda$ and of the Cox ring of $\Lambda.$

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Oskar KĘDZIERSKI. Jarosław A. WIŚNIEWSKI. "Differentials of Cox rings: Jaczewski's theorem revisited." J. Math. Soc. Japan 67 (2) 595 - 608, April, 2015. https://doi.org/10.2969/jmsj/06720595

Information

Published: April, 2015
First available in Project Euclid: 21 April 2015

zbMATH: 1336.14035
MathSciNet: MR3340188
Digital Object Identifier: 10.2969/jmsj/06720595

Subjects:
Primary: 13N05, 14E30, 14M25
Secondary: 14C20, 14F10

Rights: Copyright © 2015 Mathematical Society of Japan

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Vol.67 • No. 2 • April, 2015
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