Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T13:10:13.097Z Has data issue: false hasContentIssue false
  • English
  • Français

The Characteristic Numbers of Quartic Plane Curves

Published online by Cambridge University Press:  20 November 2018

Ravi Vakil
Ravi Vakil*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA email: vakil@math.mit.edu
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.

The characteristic numbers of smooth plane quartics are computed using intersection theory on a component of the moduli space of stable maps. This completes the verification of Zeuthen’s prediction of characteristic numbers of smooth plane curves. A short sketch of a computation of the characteristic numbers of plane cubics is also given as an illustration.

Keywords

14N1014D22
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 51 , Issue 5 , 01 October 1999 , pp. 1089 - 1120
Copyright
Copyright © Canadian Mathematical Society 1999

References

[A1] Aluffi, P., The characteristic numbers of smooth plane cubics. In: Algebraic geometry (Sundance, 1986), Lecture Notes in Math. 1311(1988), 1–8.Google Scholar
[A2] Aluffi, P., Two characteristic numbers for smooth plane curves of any degree. Trans. Amer. Math. Soc. (1) 329(1992), 73–96.Google Scholar
[AC] Aluffi, P. and Cukierman, F., Multiplicities of discriminants. Manuscripta Math. 78(1993), 245–258.Google Scholar
[DM] Deligne, P. and Mumford, D., The irreducibility of the space of curves of given genus. Inst. Hautes Études Sci. Publ. Math. 36(1969), 75–110.Google Scholar
[F] Fulton, W., Intersection Theory. Springer-Verlag, Berlin-New York, 1984.Google Scholar
[FKM] Fulton, W., Kleiman, S. and MacPherson, R., About the enumeration of contacts. In: Algebraic Geometry— Open Problems (eds. Ciliberto, C., Ghione, F. and Orecchia, F.), Lecture Notes in Math. 997(1983), 156–196.Google Scholar
[FP] Fulton, W. and Pandharipande, R., Notes on stablemaps and quantumcohomology. In: Algebraic geometry (Santa Cruz, 1995) (eds. Kollár, J., Lazarsfeld, R. and Morrison, D.), vol. 2, Amer.Math. Soc., Providence, 1997.Google Scholar
[GP] Graber, T. and Pandharipande, R., Descendant invariants and characteristic numbers in genus 0, 1 and 2. Manuscript in preparation.Google Scholar
[G] Graber, T., personal communication.Google Scholar
[HM] Harris, J. and Morrison, I., Moduli of curves. Springer-Verlag, Berlin-New York, 1998.Google Scholar
[Ha] Hartshorne, R., Algebraic geometry. Graduate Texts in Math. 52, Springer-Verlag, Berlin-New York, 1977.Google Scholar
[Hu] Hurwitz, A., Ueber Riemann’sche Flächen mit gegeben Verzweigungspunkten. Math. Ann. 39(1891), 1–61.Google Scholar
[K] Kleiman, S., Problem 15: Rigorous foundation of Schubert's enumerative calculus. In: Mathematical developments arising from Hilbert problems, Proc. Sympos. Pure Math 28(1976), 445–482.Google Scholar
[KSp] Kleiman, S. and Speiser, R., Enumerative geometry of nonsingular plane cubics. In: Algebraic geometry (Sundance, 1988), Contemp.Math. 116(1991), 85–113.Google Scholar
[S] Schubert, H., Kalkül der abzählenden Geometrie. Springer-Verlag, Berlin-New York, 1979. (“Calculus of enumerative geometry”, in German. Reprint of 1879 original, with an English introduction by S. Kleiman.)Google Scholar
[V1] Vakil, R., The enumerative geometry of rational and elliptic curves in projective space. Preprint, 1999; revised version of math.AG/9709007, available at http://www-math.mit.edu/˜vakil, submitted for publication.Google Scholar
[V2] Vakil, R., Recursions for characteristic numbers of genus one plane curves. Preprint, 1998; available at http://www-math.mit.edu/˜vakil, submitted for publication.Google Scholar
[V3] Vakil, R., Twelve points on the projective line, branched covers, and rational elliptic fibrations. Preprint, 1999; available at http://www-math.mit.edu/˜vakil.Google Scholar
[V4] Vakil, R., Characteristic numbers of rational and elliptic curves in projective space. Unpublished.Google Scholar
[vG] van Gastel, L., Characteristic numbers of plane curves: an excess intersection theoretical approach. In: Enumerative algebraic geometry (Copenhagen, 1989), Contemp.Math. 123(1991), 259–265.Google Scholar
[Z] Zeuthen, H. G., Almindelige Egenskaber ved Systemer af plane Kurver. Kongelige Danske Videnskabernes Selskabs Skrifter—Naturvidenskabelig og Mathematisk, 10(1873), 285–393. Danish with French summary.Google Scholar