Gulbrandsen, Martin G.; Halle, Lars H.; Hulek, Klaus

A GIT Construction of Degenerations of Hilbert Schemes of Points

Doc. Math. 24, 421-472 (2019)
DOI: 10.25537/dm.2019v24.421-472

Summary

We present a Geometric Invariant Theory (GIT) construction which allows us to construct good projective degenerations of Hilbert schemes of points for simple degenerations. A comparison with the construction of Li and Wu shows that our GIT stack and the stack they construct are isomorphic, as are the associated coarse moduli schemes. Our construction is sufficiently explicit to obtain good control over the geometry of the singular fibres. We illustrate this by giving a concrete description of degenerations of degree \(n\) Hilbert schemes of a simple degeneration with two components.

Mathematics Subject Classification

14D06, 14C05, 14L24, 14D23

Keywords/Phrases

geometric invariant theory, degeneration, Hilbert scheme

References

  • 1. D. Abramovich and B. Fantechi. Configurations of points on degenerate varieties and properness of moduli spaces, Rend. Semin. Mat. Univ. Padova, 137:1-17, 2017. DOI 10.4171/rsmup/137-1; zbl 1388.14040; MR3652866; arxiv 1406.2166.
  • 2. J. Alper and A. Kresch. Equivariant versal deformations of semistable curves. Mich. Math. J. 65(2):227-250, 2016. DOI 10.1307/mmj/1465329012; zbl 1346.14069; MR3510906; arxiv 1510.03201.
  • 3. J.-M. Drézet. Luna's slice theorem and applications. In Algebraic group actions and quotients, pages 39-89. Hindawi Publ. Corp., Cairo, 2004. zbl 1109.14307; MR2210794.
  • 4. M. G. Gulbrandsen, L. H. Halle, and K. Hulek. A relative Hilbert-Mumford criterion. Manuscripta Math., 148(3-4):283-301, 2015. DOI 10.1007/s00229-015-0744-8; zbl 1342.14095; MR3414477; arxiv 1404.7267.
  • 5. M. G. Gulbrandsen, L. H. Halle, K. Hulek, and Z. Zhang. The geometry of degenerations of Hilbert schemes of points. 2018. arxiv 1802.00622.
  • 6. R. Hartshorne. Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977. zbl 0367.14001; MR0463157.
  • 7. D. Huybrechts and M. Lehn. The geometry of moduli spaces of sheaves. Cambridge Mathematical Library. Cambridge University Press, Cambridge, second edition, 2010. zbl 1206.14027; MR2665168.
  • 8. J. Li. Stable morphisms to singular schemes and relative stable morphisms. J. Differential Geom., 57(3):509-578, 2001. DOI 10.4310/jdg/1090348132; zbl 1076.14540; MR1882667.
  • 9. J. Li. Good degenerations of moduli spaces. In Handbook of moduli. Vol. II, volume 25 of Adv. Lect. Math. (ALM), pages 299-351. Int. Press, Somerville, MA, 2013. zbl 1322.14009; MR3184180.
  • 10. J. Li and B. Wu. Good degeneration of quot-schemes and coherent systems. Commun. Anal. Geom. 23(4):841-921, 2015. DOI 10.4310/CAG.2015.v23.n4.a5; zbl 1349.14014; MR3385781; arxiv 1110.0390.
  • 11. D. Mumford, J. Fogarty, and F. Kirwan. Geometric invariant theory, volume 34 of Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)]. Springer-Verlag, Berlin, third edition, 1994. DOI 10.1007/978-3-642-57916-5M; zbl 0797.14004; MR1304906.
  • 12. Y. Nagai. On monodromies of a degeneration of irreducible symplectic Kähler manifolds. Math. Z., 258(2):407-426, 2008. DOI 10.1007/s00209-007-0179-3; zbl 1140.14008; MR2357645; arxiv math/0605223.
  • 13. C. P. Rourke and B. J. Sanderson. $\Delta$-sets. I. Homotopy theory. Quart. J. Math. Oxford Ser. (2), 22:321-338, 1971. DOI 10.1093/qmath/22.3.321; zbl 0226.55019; MR0300281.
  • 14. A. Vistoli. Intersection theory on algebraic stacks and on their moduli spaces. Invent. Math., 97(3):613-670, 1989. DOI 10.1007/BF01388892; zbl 0694.14001; MR1005008.
  • 15. B. Wu. A degeneration formula of Donaldson-Thomas invariants. PhD thesis, Stanford University, 2007. ProQuest LLC, Ann Arbor, MI. MR2710847.

Affiliation

Gulbrandsen, Martin G.
University of Stavanger, Department of Mathematics and Physics, 4036 Stavanger, Norway
Halle, Lars H.
University of Copenhagen, Department of Mathematical Sciences, Universitetsparken 5, 2100 Copenhagen, Denmark
Hulek, Klaus
Leibniz Universität Hannover, Institut für Algebraische Geometrie, Welfengarten 1, 30060 Hannover, Germany

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