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Bigness of the tangent
bundle of del Pezzo surfaces and $D$-simplicity
Devlin Mallory |
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Vol. 15 (2021), No. 8, 2019–2036
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Abstract |
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We consider the question of simplicity of a -algebra under the action of its ring of differential operators . We give examples to show that even when is Gorenstein and has rational singularities, need not be a simple -module; for example, this is the case when is the homogeneous coordinate ring of a smooth cubic surface. Our examples are homogeneous coordinate rings of smooth Fano varieties, and our proof proceeds by showing that the tangent bundle of such a variety need not be big. We also give a partial converse showing that when is the homogeneous coordinate ring of a smooth projective variety , embedded by some multiple of its canonical divisor, then simplicity of as a -module implies that is Fano and thus has rational singularities. |
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Keywords
$D$-simplicity, tangent bundle, bigness of tangent bundle,
positivity of vector bundles, differential operators, Fano
varieties
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Mathematical Subject Classification 2010
Primary: 13N10
Secondary: 14B05, 14J60
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Milestones
Received: 2 March 2020
Revised: 23 November 2020
Accepted: 22 December 2020
Published: 10 November 2021
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Authors |
| Devlin Mallory | |
| Department of Mathematics University of Michigan Ann Arbor, MI United States |
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