Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-26T08:43:17.241Z Has data issue: false hasContentIssue false
  • English
  • Français

Hilbert schemes of points on a locally planar curve and the Severi strata of its versal deformation

Part of: Curves

Published online by Cambridge University Press:  26 January 2012

Vivek Shende
Vivek Shende*
Affiliation:
Department of Mathematics, Princeton University, Princeton NJ, 08540, USA (email: vivek.vijay.shende@gmail.com)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let C be a locally planar curve. Its versal deformation admits a stratification by the genera of the fibres. The strata are singular; we show that their multiplicities at the central point are determined by the Euler numbers of the Hilbert schemes of points on C. These Euler numbers have made two prior appearances. First, in certain simple cases, they control the contribution of C to the Pandharipande–Thomas curve counting invariants of three-folds. In this context, our result identifies the strata multiplicities as the local contributions to the Gopakumar–Vafa BPS invariants. Second, when C is smooth away from a unique singular point, a conjecture of Oblomkov and the present author identifies the Euler numbers of the Hilbert schemes with the ‘U(∞)’ invariant of the link of the singularity. We make contact with combinatorial ideas of Jaeger, and suggest an approach to the conjecture.

MSC classification

Secondary: 14H20: Singularities, local rings
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 2 , March 2012 , pp. 531 - 547
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[Alx23]Alexander, J. W., A lemma on systems of knotted curves, Proc. Natl. Acad. 9 (1923), 93–95.CrossRefGoogle ScholarPubMed
[AK80]Altman, A. and Kleiman, S., Compactifying the Picard scheme, Adv. Math. 35 (1980), 50–112.CrossRefGoogle Scholar
[AGV88]Arnold, V., Gusein-Zade, S. and Varchenko, A., Singularities of differentiable maps, Vols. I & II (Birkhauser, 1988).CrossRefGoogle Scholar
[Beh97]Behrend, K., Gromov–Witten invariants in algebraic geometry, Invent. Math. (2) 127(3) (1997), 601–617.Google Scholar
[Beh09]Behrend, K., Donaldson–Thomas invariants via microlocal geometry, Ann. of Math. (2) 170 (2009), 1307–1338.CrossRefGoogle Scholar
[BF97]Behrend, K. and Fantechi, B., The intrinsic normal cone, Invent. Math. 128 (1997), 45–88.CrossRefGoogle Scholar
[BGS81]Briançon, J., Granger, M. and Speder, J.-P., Sur le schéma de Hilbert d’une courbe plane, Ann. Sci. Éc. Norm. Supér. (4) 14 (1981), 1–25.CrossRefGoogle Scholar
[CDG03]Campillo, A., Delgado, F. and Gusein-Zade, S. M., The Alexander polynomial of a plane curve singularity via the ring of functions on it, Duke Math J. 118 (2003), 125–156.Google Scholar
[Dem73]Demazure, M., Classification des germes à point critique isolé et à nombres de modules 0 ou 1 (d’après V. I. Arnold), Séminaire Bourbaki 443 (1973), 124–142.Google Scholar
[DH88]Diaz, S. and Harris, J., Ideals associated to deformations of singular plane curves, Trans. Amer. Math. Soc. 309 (1988), 433–468.CrossRefGoogle Scholar
[Fog68]Fogarty, J., Algebraic families on an algebraic surface, Amer. J. Math. 90 (1968), 511–521.CrossRefGoogle Scholar
[FP00]Faber, C. and Pandharipande, R., Hodge integrals and Gromov–Witten theory, Invent. Math. 139 (2000), 173–199.CrossRefGoogle Scholar
[FGS99]Fantechi, B., Göttsche, L. and van Straten, D., Euler number of the compactified Jacobian and multiplicity of rational curves, J. Algebraic Geom. 8 (1999), 115–133.Google Scholar
[GLS07]Greuel, G.-M., Lossen, C. and Shustin, E., Introduction to singularities and deformations (Springer, Berlin, 2007).Google Scholar
[GV98]Gopakumar, R. and Vafa, C., M-theory and topological strings I, arXiv:hep-th/9809187v1; M-theory and topological strings II, arXiv:hep-th/9812127.Google Scholar
[Har86]Hartshorne, R., Generalized divisors on Gorenstein curves and a theorem of Noether, J. Math. Kyoto Univ. 26 (1986), 375–386.Google Scholar
[FYHLMO85]Freyd, P., Yetter, D., Hoste, J., Lickorish, W. B. R., Millett, K. and Ocneanu, A., A new polynomial invariant of knots and links, Bull. Amer. Math. Soc. 12 (1985), 239–246.Google Scholar
[Ja91]Jaeger, F., Circuit partitions and the HOMFLY polynomial of closed braids, Trans. Amer. Math. Soc. 323 (1991), 449–463.CrossRefGoogle Scholar
[KKV99]Katz, S., Klemm, A. and Vafa, C., M-theory, topological strings, and spinning black holes, arXiv:hep-th/9910181v2.Google Scholar
[Kaw03]Kawai, T., String and vortex, arXiv:hep-th/0312243v4.Google Scholar
[LT98]Li, J. and Tian, G., Virtual moduli cycles and Gromov–Witten invariants of algebraic varieties, J. Amer. Math. Soc. 11 (1998), 119–174.CrossRefGoogle Scholar
[Mac62]Macdonald, I. G., Symmetric products of an algebraic curve, Topology 1 (1962), 319–343.Google Scholar
[MNOP06]Maulik, D., Nekrasov, N., Okounkov, A. and Pandharipande, R., Gromov–Witten theory and Donaldson–Thomas theory I, Compositio Math. 142 (2006), 1263–1285.CrossRefGoogle Scholar
[Mil68]Milnor, J., Singular points of complex hypersurfaces (Princeton University Press, Princeton, NJ, 1968).Google Scholar
[Moi81]Moishezon, B. G., Stable branch curves and braid monodromies, Lecture Notes in Mathematics, vol. 862 (Springer, New York, 1981), 107–192.Google Scholar
[OS10]Oblomkov, A. and Shende, V., The Hilbert scheme of a plane curve singularity, and the HOMFLY polynomial of its link, Duke Math. J., to appear, arXiv:1003.1568v1 [math.AG].Google Scholar
[PT09]Pandharipande, R. and Thomas, R., Curve counting via stable pairs in the derived category, Invent. Math. 178 (2009), 407–477.CrossRefGoogle Scholar
[PT10]Pandharipande, R. and Thomas, R. P., Stable pairs and BPS invariants, J. Amer. Math. Soc. 23 (2010), 267–297.CrossRefGoogle Scholar
[Tes80]Tessier, B., Résolution simultanée - I. Famille de courbes, in Séminaire sur les singularités des surfaces, Lecture Notes in Mathematics, vol. 777 (Springer, New York, 1980).Google Scholar
[Wal04]Wall, C. T. C., Singular points of plane curves (Cambridge University Press, Cambridge, 2004).CrossRefGoogle Scholar
[Yam84]Yamamoto, M., Classification of isolated algebraic singularities by their Alexander polynomials, Topology 23 (1984), 277–287.CrossRefGoogle Scholar
[Zar71]Zariski, O., General theory of saturation and of saturated local rings II: saturated local rings of dimension 1, Amer. J. Math. 93 (1971), 872–964.CrossRefGoogle Scholar
[ZT86]Zariski, O. and Tessier, B., Le problème des modules pour les branches planes (Hermann, Paris, 1986).Google Scholar