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2018 Rational singularities of $G$-saturation
Nham V. Ngo
J. Commut. Algebra 10(3): 375-391 (2018). DOI: 10.1216/JCA-2018-10-3-375

Abstract

Let $G$ be a semisimple algebraic group defined over an algebraically closed field of characteristic 0 and $P$ a parabolic subgroup of $G$. Let $M$ be a $P$-module and $V$ a $P$-stable closed subvariety of $M$. We show in this paper that, if the varieties $V$ and $G\cdot M$ have rational singularities, and the induction functor $R^i ind _P^G(-)$ satisfies certain vanishing conditions, then the variety $G\cdot V$ has rational singularities. This generalizes a result of Kempf on the collapsing of homogeneous bundles. As an application, we prove the property of having rational singularities for nilpotent commuting varieties over $3\times 3$ matrices.

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Nham V. Ngo. "Rational singularities of $G$-saturation." J. Commut. Algebra 10 (3) 375 - 391, 2018. https://doi.org/10.1216/JCA-2018-10-3-375

Information

Published: 2018
First available in Project Euclid: 9 November 2018

zbMATH: 06976322
MathSciNet: MR3874659
Digital Object Identifier: 10.1216/JCA-2018-10-3-375

Subjects:
Primary: 13A50 , 14L30 , 14M20

Keywords: $G$-saturation varieties , algebraic groups , Cohomology , commuting varieties , homogeneous bundles , Rational resolution (singularities)

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.10 • No. 3 • 2018
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