A relative Hilbert-Mumford criterion

Martin G. Gulbrandsen; Lars Halvard Halle; Klaus Hulek

doi:10.1007/s00229-015-0744-8

A relative Hilbert-Mumford criterion

Martin G. Gulbrandsen, Lars Halvard Halle, Klaus Hulek

Department of Mathematical Sciences

6 Citations (Scopus)

Abstract

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→ SpecA to a noetherian k-algebra A. We also extend the classical projectivity result for GIT quotients: the induced morphism (Formula presented.) SpecAG is projective. As an example of applications to moduli problems, we consider degenerations of Hilbert schemes of points.

Translated title of the contribution	A relative Hilbert-Mumford criterion
Original language	English
Journal	Manuscripta Mathematica
Volume	148
Issue number	3
Pages (from-to)	283-301
ISSN	0025-2611
DOIs	https://doi.org/10.1007/s00229-015-0744-8
Publication status	Published - 1 Nov 2015

Keywords

math.AG
14L24 (Primary) 13A50, 14D06 (Secondary)

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10.1007/s00229-015-0744-8

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AU - Gulbrandsen, Martin G.

AU - Halle, Lars Halvard

AU - Hulek, Klaus

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Y1 - 2015/11/1

N2 - 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→ SpecA to a noetherian k-algebra A. We also extend the classical projectivity result for GIT quotients: the induced morphism (Formula presented.) SpecAG is projective. As an example of applications to moduli problems, we consider degenerations of Hilbert schemes of points.

AB - 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→ SpecA to a noetherian k-algebra A. We also extend the classical projectivity result for GIT quotients: the induced morphism (Formula presented.) SpecAG is projective. As an example of applications to moduli problems, we consider degenerations of Hilbert schemes of points.

KW - math.AG

KW - 14L24 (Primary) 13A50, 14D06 (Secondary)

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DO - 10.1007/s00229-015-0744-8

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SP - 283

EP - 301

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

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ER -

A relative Hilbert-Mumford criterion

Abstract

Keywords

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