Vol. 15, No. 8, 2021

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Bigness of the tangent bundle of del Pezzo surfaces and $D$-simplicity

Devlin Mallory

Vol. 15 (2021), No. 8, 2019–2036
Abstract

We consider the question of simplicity of a -algebra R under the action of its ring of differential operators DR. We give examples to show that even when R is Gorenstein and has rational singularities, R need not be a simple DR-module; for example, this is the case when R 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 R is the homogeneous coordinate ring of a smooth projective variety X, embedded by some multiple of its canonical divisor, then simplicity of R as a DR-module implies that X is Fano and thus R has rational singularities.

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Keywords
$D$-simplicity, tangent bundle, bigness of tangent bundle, positivity of vector bundles, differential operators, Fano varieties
Mathematical Subject Classification 2010
Primary: 13N10
Secondary: 14B05, 14J60
Milestones
Received: 2 March 2020
Revised: 23 November 2020
Accepted: 22 December 2020
Published: 10 November 2021
Authors
Devlin Mallory
Department of Mathematics
University of Michigan
Ann Arbor, MI
United States