A relative Hilbert–Mumford criterion

We generalize the classical Hilbert–Mumford criteria for GIT (semi-)stability in terms of one parameter subgroups of a linearly reductive group G over a field k , to the relative situation of an equivariant, projective morphism X → Spec A to a noetherian k -algebra A . We also extend the classical p...

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Bibliographic Details
Published in:Manuscripta mathematica 2015-11, Vol.148 (3-4), p.283-301
Main Authors: Gulbrandsen, Martin G., Halle, Lars H., Hulek, Klaus
Format: Article
Language:English
Subjects:
Algebraic Geometry
Calculus of Variations and Optimal Control
Optimization
Geometry
Lie Groups
Mathematics
Mathematics and Statistics
Number Theory
Topological Groups
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Summary:We generalize the classical Hilbert–Mumford criteria for GIT (semi-)stability in terms of one parameter subgroups of a linearly reductive group G over a field k , to the relative situation of an equivariant, projective morphism X → Spec A to a noetherian k -algebra A . We also extend the classical projectivity result for GIT quotients: the induced morphism X s s / G → Spec A G is projective. As an example of applications to moduli problems, we consider degenerations of Hilbert schemes of points.
ISSN:0025-2611
1432-1785
DOI:10.1007/s00229-015-0744-8