Journal of the Mathematical Society of Japan

Differentials of Cox rings: Jaczewski's theorem revisited


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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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J. Math. Soc. Japan, Volume 67, Number 2 (2015), 595-608.

First available in Project Euclid: 21 April 2015

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Primary: 14M25: Toric varieties, Newton polyhedra [See also 52B20] 13N05: Modules of differentials 14E30: Minimal model program (Mori theory, extremal rays)
Secondary: 14C20: Divisors, linear systems, invertible sheaves 14F10: Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials [See also 13Nxx, 32C38]

Cox ring Mori Dream Space Euler sequence differentials toric varieties


KĘDZIERSKI, Oskar; WIŚNIEWSKI, Jarosław A. Differentials of Cox rings: Jaczewski's theorem revisited. J. Math. Soc. Japan 67 (2015), no. 2, 595--608. doi:10.2969/jmsj/06720595.

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