The Characteristic Numbers of Quartic Plane Curves
Canadian journal of mathematics, Tome 51 (1999) no. 5, pp. 1089-1120

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CJM-1999-048-1
Mots-clés : 14N10, 14D22
Vakil, Ravi. The Characteristic Numbers of Quartic Plane Curves. Canadian journal of mathematics, Tome 51 (1999) no. 5, pp. 1089-1120. doi: 10.4153/CJM-1999-048-1
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[A1] [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] [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] [AC] Aluffi, P. and Cukierman, F., Multiplicities of discriminants. Manuscripta Math. 78(1993), 245–258. Google Scholar

[DM] [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] [F] Fulton, W., Intersection Theory. Springer-Verlag, Berlin-New York, 1984. Google Scholar

[FKM] [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] [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] [GP] Graber, T. and Pandharipande, R., Descendant invariants and characteristic numbers in genus 0, 1 and 2. Manuscript in preparation. Google Scholar

[G] [G] Graber, T., personal communication. Google Scholar

[HM] [HM] Harris, J. and Morrison, I., Moduli of curves. Springer-Verlag, Berlin-New York, 1998. Google Scholar

[Ha] [Ha] Hartshorne, R., Algebraic geometry. Graduate Texts in Math. 52, Springer-Verlag, Berlin-New York, 1977. Google Scholar

[Hu] [Hu] Hurwitz, A., Ueber Riemann’sche Flächen mit gegeben Verzweigungspunkten. Math. Ann. 39(1891), 1–61. Google Scholar

[K] [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] [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] [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] [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] [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] [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] [V4] Vakil, R., Characteristic numbers of rational and elliptic curves in projective space. Unpublished. Google Scholar

[vG] [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] [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

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