Abstract
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.
Citation
Devlin Mallory. "Bigness of the tangent bundle of del Pezzo surfaces and -simplicity." Algebra Number Theory 15 (8) 2019 - 2036, 2021. https://doi.org/10.2140/ant.2021.15.2019
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