Manuscripta Mathematica

, Volume 148, Issue 3–4, pp 283–301 | Cite as

A relative Hilbert–Mumford criterion

  • Martin G. Gulbrandsen
  • Lars H. Halle
  • Klaus Hulek


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 \({X \rightarrow \,{\rm Spec}\,\, A}\) to a noetherian k-algebra A. We also extend the classical projectivity result for GIT quotients: the induced morphism \({X^{ss} /G \rightarrow \,{\rm Spec}\,\, A^G}\) is projective. As an example of applications to moduli problems, we consider degenerations of Hilbert schemes of points.

Mathematics Subject Classification

14L24 (primary) 13A50 14D06 (secondary) 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Martin G. Gulbrandsen
    • 1
  • Lars H. Halle
    • 2
  • Klaus Hulek
    • 3
    • 4
  1. 1.Department of Mathematics and Natural SciencesUniversity of StavangerStavangerNorway
  2. 2.Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark
  3. 3.Institut für Algebraische GeometrieLeibniz Universität HannoverHannoverGermany
  4. 4.Institute for Advanced StudySchool of MathematicsPrincetonUSA

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