Skip to main content
SpringerLink
Log in

An example of an SL2-Hilbert scheme with multiplicities

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

In [AB05], Alexeev and Brion have introduced the notion of invariant Hilbert schemes. We determine the invariant Hilbert scheme of the zero fibre of the moment map of an action of SL2 on \( {\left( {{\mathbb{C}^2}} \right)^{ \oplus 6}} \) as one of the first examples of invariant Hilbert schemes with multiplicities. While doing this, we present a general procedure for realizing these calculations. We also consider questions of smoothness and connectedness and thereby show that our Hilbert scheme gives a resolution of singularities of the symplectic reduction of the action.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. V. Alexeev, M. Brion, Stable reductive varieties. I. Affine varieties, Invent. Math. 157 (2004), no. 2, 227–274.

    Article  MathSciNet  MATH  Google Scholar 

  2. V. Alexeev, M. Brion, Moduli of affine schemes with reductive group action, J. Algebraic Geom. 14 (2005), no. 1, 83–117.

    Article  MathSciNet  MATH  Google Scholar 

  3. P. Bravi, S. Cupit-Foutou, Equivariant deformations of the affine multicone over a flag variety, Adv. Math. 217 (2008), no. 6, 2800–2821.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Beauville, Variétés Kähleriennes dont la première classe de Chern est nulle, J. Differential Geom. 18 (1983), no. 4, 755–782.

    MathSciNet  MATH  Google Scholar 

  5. T. Becker, On the existence of symplectic resolutions of symplectic reductions, Math. Z. 265 (2010), no. 2, 343–363.

    Article  MathSciNet  MATH  Google Scholar 

  6. T. Bridgeland, A. King, M. Reid, The McKay correspondence as an equivalence of derived categories, J. Amer. Math. Soc. 14 (2001), no. 3, 535–554.

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Brion, Invariant Hilbert schemes, arXiv:1102.0198v2 (2011).

  8. J. Budmiger, Deformation of orbits in minimal sheets, Dissertation, Universität Basel, 2010.

  9. I. Dolgachev, Lectures on Invariant Theory, London Mathematical Society Lecture Note Series, Vol. 296, Cambridge University Press, Cambridge, 2003.

    Google Scholar 

  10. J. Fogarty, Algebraic families on an algebraic surface, Amer. J. Math 90 (1968), 511–521.

    Article  MathSciNet  MATH  Google Scholar 

  11. D. Mumford, J. Fogarty, F. Kirwan. Geometric Invariant Theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 34, Springer-Verlag, Berlin, 1994.

  12. R. Howe, Perspectives on invariant theory: Schur duality, multiplicity-free actions and beyond, in: The Schur Lectures, (Tel Aviv, 1992), Israel Math. Conf. Proc., Vol. 8, Bar-Ilan University, Ramat Gan, 1995, pp. 1–182.

  13. M. Haiman, B. Sturmfels, Multigraded Hilbert schemes, J. Algebraic Geom. 13 (2004), no. 4, 725–769.

    Article  MathSciNet  MATH  Google Scholar 

  14. Y. Ito, I. Nakamura, McKay correspondence and Hilbert schemes, Proc. Japan Acad. Ser. A Math. Sci. 72 (1996), no. 7, 135–138.

    Article  MathSciNet  MATH  Google Scholar 

  15. Y. Ito, I. Nakamura, Hilbert schemes and simple singularities, in: New Trends in Algebraic Geometry, (Warwick, 1996), London Math. Soc. Lecture Note Ser., Vol. 264, Cambridge University Press, Cambridge, 1999, pp. 151–233.

  16. S. Jansou, Déformations des cônes de vecteurs primitifs, Math. Ann. 338 (2007), no. 3, 627–667.

    Article  MathSciNet  MATH  Google Scholar 

  17. S. Jansou, N. Ressayre, Invariant deformations of orbit closures in \( \mathfrak{s}{\mathfrak{l}_n} \), Represent. Theory 13 (2009), 50–62.

    Article  MathSciNet  MATH  Google Scholar 

  18. S. Papadakis, B. van Steirteghem, Equivariant deformations of spherical modules for groups of type A, arXiv:1008.0911v2 (2010).

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Johannes Gutenberg-Universität, Staudingerweg 9, 55099, Mainz, Germany

    Tanja Becker

Authors
  1. Tanja Becker
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tanja Becker.

Additional information

Partially supported by DAAD and SFB/TR 45 “Periods, Moduli Spaces and Arithmetic of Algebraic Varieties” of the DFG.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Becker, T. An example of an SL2-Hilbert scheme with multiplicities. Transformation Groups 16, 915–938 (2011). https://doi.org/10.1007/s00031-011-9153-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-011-9153-6

Keywords

Search

Navigation