Inventiones mathematicae

, Volume 191, Issue 3, pp 671–718

Finite Hilbert stability of (bi)canonical curves

  • Jarod Alper
  • Maksym Fedorchuk
  • David Ishii Smyth
Article

Abstract

We prove that a generic canonically or bicanonically embedded smooth curve has semistable mth Hilbert points for all m≥2. We also prove that a generic bicanonically embedded smooth curve has stable mth Hilbert points for all m≥3. In the canonical case, this is accomplished by proving finite Hilbert semistability of special singular curves with \(\mathbb{G}_{m}\)-action, namely the canonically embedded balanced ribbon and the canonically embedded balanced doubleA2k+1-curve. In the bicanonical case, we prove finite Hilbert stability of special hyperelliptic curves, namely Wiman curves. Finally, we give examples of canonically embedded smooth curves whose mth Hilbert points are non-semistable for low values of m, but become semistable past a definite threshold.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alper, J., Fedorchuk, M., Smyth, D.I.: Singularities with \(\mathbb{G}_{m}\)-action and the log minimal model program for \(\overline{M}_{g}\) (2010). arXiv:1010.3751v2 [math.AG]
  2. 2.
    Barja, M.A.: On the slope of bielliptic fibrations. Proc. Am. Math. Soc. 129(7), 1899–1906 (2001) (electronic) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bayer, D., Eisenbud, D.: Ribbons and their canonical embeddings. Trans. Am. Math. Soc. 347(3), 719–756 (1995) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Barth, W.P., Hulek, K., Peters, C.A.M., Van de Ven, A.: Compact Complex Surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 4, 2nd edn. Springer, Berlin (2004) MATHGoogle Scholar
  5. 5.
    Cornalba, M., Harris, J.: Divisor classes associated to families of stable varieties, with applications to the moduli space of curves. Ann. Sci. Ec. Norm. Super. (4) 21(3), 455–475 (1988) MathSciNetMATHGoogle Scholar
  6. 6.
    Fedorchuk, M.: The final log canonical model of the moduli space of stable curves of genus 4. Int. Math. Res. Not. (2012). doi:10.1093/imrn/rnr242 Google Scholar
  7. 7.
    Fedorchuk, M., Jensen, D.: Stability of 2nd Hilbert points of canonical curves. Int. Math. Res. Not. (in press). arXiv:1111.5339v2 [math.AG]
  8. 8.
    Fong, L.-Y.: Rational ribbons and deformation of hyperelliptic curves. J. Algebr. Geom. 2(2), 295–307 (1993) MathSciNetMATHGoogle Scholar
  9. 9.
    Fedorchuk, M., Smyth, D.I.: Alternate compactifications of moduli spaces of curves. In: Farkas, G., Morrison, I. (eds.) Handbook of Moduli (in press). arXiv:1012.0329v2 [math.AG]
  10. 10.
    Gieseker, D.: Lectures on Moduli of Curves. Tata Institute of Fundamental Research Lectures on Mathematics and Physics, vol. 69. Tata Institute of Fundamental Research, Bombay (1982) MATHGoogle Scholar
  11. 11.
    Gieseker, D.: Geometric invariant theory and applications to moduli problems. In: Invariant Theory. Proceedings of the 1st 1982 Session of the Centro Internazionale Matematico Estivo (CIME), Montecatini, June 10–18, 1982. Lecture Notes in Mathematics, vol. 996, pages v+159. Springer, Berlin (1983) Google Scholar
  12. 12.
    Harris, J.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 133. Springer, New York (1992). A first course CrossRefGoogle Scholar
  13. 13.
    Hassett, B., Hyeon, D.: Log minimal model program for the moduli space of curves: the first flip (2008). arXiv:0806.3444 [math.AG]
  14. 14.
    Hassett, B., Hyeon, D.: Log canonical models for the moduli space of curves: the first divisorial contraction. Trans. Am. Math. Soc. 361(8), 4471–4489 (2009) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Hyeon, D., Lee, Y.: Log minimal model program for the moduli space of stable curves of genus three. Math. Res. Lett. 17(4), 625–636 (2010) MathSciNetMATHGoogle Scholar
  16. 16.
    Hyeon, D., Morrison, I.: Stability of tails and 4-canonical models. Math. Res. Lett. 17(4), 721–729 (2010) MathSciNetMATHGoogle Scholar
  17. 17.
    Kempf, G.R.: Instability in invariant theory. Ann. Math. (2) 108(2), 299–316 (1978) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Kollár, J.: Rational Curves on Algebraic Varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 32 [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Berlin (1996) CrossRefGoogle Scholar
  19. 19.
    Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete (2), vol. 34, 3rd edn. [Results in Mathematics and Related Areas (2)]. Springer, Berlin (1994) CrossRefGoogle Scholar
  20. 20.
    Morrison, I.: GIT constructions of moduli spaces of stable curves and maps. In: Surveys in Differential Geometry. Vol. XIV. Geometry of Riemann Surfaces and Their Moduli Spaces, vol. 14, pp. 315–369. Int. Press, Somerville (2009) Google Scholar
  21. 21.
    Morrison, I., Swinarski, D.: Groebner techniques for low degree Hilbert stability. Exp. Math. 20(1), 34–56 (2011) MathSciNetCrossRefGoogle Scholar
  22. 22.
    Mumford, D.: Stability of projective varieties. Enseign. Math. (2) 23(1–2), 39–110 (1977) MathSciNetMATHGoogle Scholar
  23. 23.
    Schubert, D.: A new compactification of the moduli space of curves. Compos. Math. 78(3), 297–313 (1991) MathSciNetMATHGoogle Scholar
  24. 24.
    Serre, J.-P.: Algebraic Groups and Class Fields. Graduate Texts in Mathematics, vol. 117. Springer, New York (1988). Translated from the French MATHCrossRefGoogle Scholar
  25. 25.
    Stankova-Frenkel, Z.-E.: Moduli of trigonal curves. J. Algebr. Geom. 9(4), 607–662 (2000) MathSciNetMATHGoogle Scholar
  26. 26.
    van der Wyck, F.: Moduli of singular curves and crimping. Ph.D. thesis, Harvard University (2010) Google Scholar
  27. 27.
    Wiman, A.: Über die Doppelcurve auf den geradlinigen Flächen. Acta Math. 19(1), 63–71 (1895) MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Xiao, G.: Fibered algebraic surfaces with low slope. Math. Ann. 276(3), 449–466 (1987) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Jarod Alper
    • 1
  • Maksym Fedorchuk
    • 2
  • David Ishii Smyth
    • 3
  1. 1.Departamento de MatemáticasUniversidad de los AndesBogotáColombia
  2. 2.Department of MathematicsColumbia UniversityNew YorkUSA
  3. 3.Department of MathematicsHarvard UniversityCambridgeUSA

Personalised recommendations