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Observability in invariant theory

Author: Lex E. Renner
Journal: Proc. Amer. Math. Soc. 141 (2013), 205-216
MSC (2010): Primary 13A50, 14L30
Published electronically: May 24, 2012
MathSciNet review: 2988723
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Abstract: We consider actions $ G\times X\to X$ of the affine, algebraic group $ G$ on the affine, algebraic variety $ X$. We say that $ G\times X\to X$ is observable in codimension one if for any height-one, $ G$-invariant, prime ideal $ \mathfrak{p}\subset k[X]$, $ \mathfrak{p}^G\neq (0)$. Many familiar actions are observable in codimension one. We characterize such actions geometrically and indicate how they fit into the general framework of invariant theory. We look at what happens if we impose further restrictions, such as $ G$ being reductive or $ X$ being factorial. We indicate how Grosshans subgroups are involved.

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Additional Information

Lex E. Renner
Affiliation: Department of Mathematics, University of Western Ontario, London, ON, N6A 5B7, Canada

Received by editor(s): March 8, 2011
Received by editor(s) in revised form: May 31, 2011, and June 22, 2011
Published electronically: May 24, 2012
Communicated by: Harm Derksen
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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