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
Article

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 \({X \rightarrow \,{\rm Spec}\,\, A}\) 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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References

  1. 1.
    Abramovich D., Cadman C., Fantechi B., Wise J.: Expanded degenerations and pairs. Commun. Algebra 41(6), 2346–2386 (2013)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Hu Y.: Relative geometric invariant theory and universal moduli spaces. Int. J. Math. 7(2), 151–181 (1996)MATHCrossRefGoogle Scholar
  3. 3.
    Huybrechts, D., Lehn, M.: The Geometry of Moduli Spaces of Sheaves. Aspects of Mathematics, vol. E31. Friedr. Vieweg & Sohn, Berlin (1997)Google Scholar
  4. 4.
    King A.D.: Moduli of representations of finite-dimensional algebras. Q. J. Math. Oxf. Ser. (2) 45(180), 515–530 (1994)MATHCrossRefGoogle Scholar
  5. 5.
    Li J.: Stable morphisms to singular schemes and relative stable morphisms. J. Differ. Geom. 57(3), 509–578 (2001)MATHGoogle Scholar
  6. 6.
    Li, J., Wu, B.: Good degenerations of Quot-schemes and coherent systems (2011) (preprint). arXiv:1110.0390
  7. 7.
    Liu Q.: Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics, vol. 6. Oxford University Press, Oxford (2002)Google Scholar
  8. 8.
    Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory, 3rd edn. Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer, Berlin (1994)Google Scholar
  9. 9.
    Newstead, P.: Introduction to Moduli Problems and Orbit Spaces. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 51. Tata Institute of Fundamental Research, Bombay; by the Narosa Publishing House, New Delhi (1978)Google Scholar
  10. 10.
    Reichstein Z.: Stability and equivariant maps. Invent. Math. 96(2), 349–383 (1989)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Schmitt A.: Global boundedness for decorated sheaves. Int. Math. Res. Not. 68, 3637–3671 (2004)CrossRefGoogle Scholar
  12. 12.
    Seshadri C.S.: Quotient spaces modulo reductive algebraic groups. Ann. Math. (2) 95(95), 511–556 (1972)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Seshadri C.S.: Geometric reductivity over arbitrary base. Adv. Math. 26(3), 225–274 (1977)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    The Stacks Project Authors, Stacks Project. http://stacks.math.columbia.edu (2014)

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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