Advertisement

manuscripta mathematica

, Volume 151, Issue 3–4, pp 289–304 | Cite as

On Chow stability for algebraic curves

  • L. Brambila-PazEmail author
  • H. Torres-López
Article
  • 98 Downloads

Abstract

In the last decades there have been introduced different concepts of stability for projective varieties. In this paper we give a natural and intrinsic criterion for the Chow and Hilbert stability for complex irreducible smooth projective curves \(C\subset {\mathbb {P}} ^n\). Namely, if the restriction of the tangent bundle of \({\mathbb {P}} ^n\) to C is stable then \(C\subset {\mathbb {P}} ^n\) is Chow stable, and hence Hilbert stable. We apply this criterion to describe a smooth open set of a regular component of the locus of Chow stable curves of the Hilbert scheme of \(\mathbb {P} ^n\) with Hilbert polynomial \(P(t)=dt+(1-g)\), when \(g\ge 4\) and \(d>g+n-\left\lfloor \frac{g}{n+1}\right\rfloor .\) Moreover, we describe the quotient stack of such curves. Similar results are obtained for the locus of Hilbert stable curves.

Mathematics Subject Classification

14H60 14H10 14C05 14D23 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alexeev, V.: Higher-dimensional analogues of stable curves. In: Sanz, M., Soria, J., Varona, J., Verdera, J. (eds.) Algebraic and Complex Geometry, pp. 515–536. Soc. Publ. House, Proceedings of Madrid ICM, European Math (2006)Google Scholar
  2. 2.
    Alper, J., Hyeon, D.: GIT construction of log canonical models of \(\bar{M_g}\). In: Alexeev, V., Gibney, A., Izadi, E., Kollár, J., Looijenga, E. (eds.) Compact Moduli Spaces and Vector Bundles. Contemporary Mathematicas, vol. 564, pp. 87–106. American Mathematics Society, Providence RI (2012)CrossRefGoogle Scholar
  3. 3.
    Arbarello, E., Cornalba, M., Griffiths, P.A.: Geometry of Algebraic Curves, with a Contribution by Joseph Daniel Harris. Vol. II, Volume 268 of Grundlehren der Mathematischen Wissenschaften. Springer, Heidelberg (2011)Google Scholar
  4. 4.
    Barja, M.A., Stoppino, L.: Stability conditions and positivity of invariants of fibrations. In: Frübis-Krüger, A., Nanne, R., Schütt, M. (eds.) Algebraic and Complex Geometry, in Honour of Klaus Huleks 60th Birthday, vol. 71, pp. 1–40. Springer Proc. Math. Stat. (2014)Google Scholar
  5. 5.
    Berman, J.: K-polystability of Q-Fano varieties admitting Kahler–Einstein metrics. Invent. Math. 203(3), 973–1025 (2015). doi: 10.1007/s00222-015-0607-7
  6. 6.
    Bhosle, U.N., Brambila-Paz, L., Newstead, P.E.: On coherent systems of type \((n, d, n + 1)\) on Petri curves. Manuscr. Math. 126, 409–441 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bhosle, U.N., Brambila-Paz, L., Newstead, P.E.: On linear series and a conjecture of DC Butler. Int. J. Math. 26(2), 1550007 (2015). doi: 10.1142/S0129167X1550007X MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bini, G., Felici, F., Melo, M., Viviani, F.: GIT for polarized curves. In: Lecture Notes in Mathematics, 2122. Springer, Berlin (2014)Google Scholar
  9. 9.
    Brambila-Paz, L.: Non-emptiness of moduli spaces of coherent systems. Int. J. Math. 19(7), 779–799 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Brambila-Paz, L., Ortega, A.: Brill–Noether bundles and coherent systems on special curves. In: Brambila-Paz, L., Steven, B., Garcia-Prada, O., Ramanan, S. (eds.) Moduli Spaces and Vector Bundles. LMS Lecture Note Series 359, pp. 456–472. Cambridge University Press, Cambridge (2009)Google Scholar
  11. 11.
    Butler, D.C.: Normal generation of vector bundles over a curve. J. Differ. Geom. 39(1), 1–34 (1994)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Butler, D.C.: Birational maps of moduli of Brill–Noether pairs. Preprint arXiv:alg-geom/9705009
  13. 13.
    Camere, C.: About the stability of the tangent bundle of \({\mathbb{P}} ^n\) restricted to a curve. C. R. Math. Acad. Sci. Paris Ser I 346, 421–426 (2008)Google Scholar
  14. 14.
    Chen, X.-X., Donaldson, S., Sun, S.: Kähler-Einstein metrics and stability. Int. Math. Res. Not. IMRN 8, 2119–2125 (2014)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Ein, L., Lazarsfeld, R.: Stability and restrictions of Picard bundles, with an application to thenormal bundles of elliptic curves. In: Ellingsrud, G., Peskine, C., Sacchiero, G., Stromme, S.A. (eds.) Complex Projective Geometry (Trieste 1989/Bergen 1989). LMS Lecture Note Series, vol. 179, pp. 149–156. Cambridge University Press, Cambridge (1992)CrossRefGoogle Scholar
  16. 16.
    Fantechi, B., Göttsche, L., Illusie, L., Kleiman, S.L., Nitsure, N., Vistolli, A.: Fundametal Algebraic Geometry: Grothendieck’s FGA Explained. Mathematical Surveys and Monographs, AMS 123, (2005)Google Scholar
  17. 17.
    Gieseker, D.: Lectures on Moduli of Curves. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Volume 69, Bombay (1982)Google Scholar
  18. 18.
    Grothendieck, A.: Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné): III. Étude cohomologique des faisceaux cohérents, Premiére partie. Publications Mathematiques de l’IHÉS 11, 5–167 (1961)Google Scholar
  19. 19.
    Harris, J., Morrison, I.: Moduli of curves. In: Graduate Texts in Mathematics, vol. 187 (1998)Google Scholar
  20. 20.
    Hassett, B., Hyeon, D.: Log minimal model program for the moduli space of curves: the first flip. Ann. Math. (2) 177(3), 911–968 (2013)Google Scholar
  21. 21.
    Kollär, J.: Moduli of varieties of general type. In: Handbook of Moduli. Adv. Lect. Math. (ALM), 25, Vol. II, pp. 131–157. Int. Press, Somerville (2013)Google Scholar
  22. 22.
    Lange, H., Newstead, P.E.: Coherent systems on elliptic curves. Int. J. Math. 16, 787–805 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mabuchi, T.: Chow-stability and Hilbert-stability in Mumford’s geometric invariant theory. Osaka J. Math. 45, 833–846 (2008)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Mistretta, E.C.: Stability of line bundle transforms on curves with respect to low codimensional subspaces. J. Lond. Math. Soc. (2) 78(1), 172–182 (2008)Google Scholar
  25. 25.
    Mistretta, E.C., Stoppino, L.: Linear series on curves: stability and Clifford index. Int. J. Math. 23(12) (2012). doi: 10.1142/0129167X12501212
  26. 26.
    Morrison, I.: Projective stability of ruled surfaces. Invent. Math. 56, 269–304 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mumford, D.: Stability of projective varieties. Enseign. Math. 2(23), 39–110 (1977)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Odaka, Y.: On the GIT stability of polarized varieties a survey. In: Proceedings of Kinosaki Algebraic Geometry symposium (2010)Google Scholar
  29. 29.
    Paranjape, K., Ramanan, S.: On the canonical ring of a curve. In: Algebraic Geometry and Commutative Algebra, in Honor of Masayoshi Nagata, vol. 2, pp. 503–516, Kinokuniya (1987)Google Scholar
  30. 30.
    Ross, J., Thomas, R.: A study of the Hilbert Mumford criterion for the stability of projective varieties. J. Algebr. Geom. 16, 201–255 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Schneider, O.: Stabilité des fibrés \(\wedge ^p E_L\) et condition de Raynaud. Fac. Sci. Toulouse Math. (6) 14(3), 515–525 (2005)Google Scholar
  32. 32.
    Tian, G.: Kähler-Einstein metrics with positive scalar curvature. Invent. Math. 130, 1–39 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Centro de Investigación en MatemáticasGuanajuatoMexico

Personalised recommendations