manuscripta mathematica

, Volume 102, Issue 1, pp 53–84

Counting curves on rational surfaces

  • Ravi Vakil

DOI: 10.1007/s002291020053

Cite this article as:
Vakil, R. manuscripta math. (2000) 102: 53. doi:10.1007/s002291020053


In [CH3], Caporaso and Harris derive recursive formulas counting nodal plane curves of degree d and geometric genus g in the plane (through the appropriate number of fixed general points). We rephrase their arguments in the language of maps, and extend them to other rational surfaces, and other specified intersections with a divisor. As applications, (i) we count irreducible curves on Hirzebruch surfaces in a fixed divisor class and of fixed geometric genus, (ii) we compute the higher-genus Gromov–Witten invariants of (or equivalently, counting curves of any genus and divisor class on) del Pezzo surfaces of degree at least 3. In the case of the cubic surface in (ii), we first use a result of Graber to enumeratively interpret higher-genus Gromov–Witten invariants of certain K-nef surfaces, and then apply this to a degeneration of a cubic surface.

Mathematics Subject Classification (1991):14N10, 14H10, 14J10 

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Ravi Vakil
    • 1
  1. 1.M.I.T. Department of Mathematics, 77 Massachusetts Ave. Rm. 2-271, Cambridge, MA 02139, USA. e-mail: