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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, 14E15, 14E05, 12E05, 20G10 show english descriptions Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17]
Global theory and resolution of singularities [See also 14B05, 32S20, 32S45]
Rational and birational maps
Polynomials (irreducibility, etc.)
Cohomology theory
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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