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Essential Dimensions of Algebraic Groups and a Resolution Theorem for $G$-Varieties

Published:2000-10-01
Printed: Oct 2000
• Zinovy Reichstein
• Boris Youssin
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Abstract

Let $G$ be an algebraic group and let $X$ be a generically free $G$-variety. We show that $X$ can be transformed, by a sequence of blowups with smooth $G$-equivariant centers, into a $G$-variety $X'$ with the following property the stabilizer of every point of $X'$ is isomorphic to a semidirect product $U \sdp A$ of a unipotent group $U$ and a diagonalizable group $A$. As an application of this result, we prove new lower bounds on essential dimensions of some algebraic groups. We also show that certain polynomials in one variable cannot be simplified by a Tschirnhaus transformation.
 MSC Classifications: 14L30 - Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17] 14E15 - Global theory and resolution of singularities [See also 14B05, 32S20, 32S45] 14E05 - Rational and birational maps 12E05 - Polynomials (irreducibility, etc.) 20G10 - Cohomology theory

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