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Enumerating Rational Curves: The Rational Fibration Method

Published online by Cambridge University Press:  04 December 2007

LUCIA CAPORASO  and
JOE HARRIS
LUCIA CAPORASO
Affiliation:
Mathematics Department, Harvard University, Cambridge MA 02138, USA and Dipartimento di Matematica, Universitá di Roma ‘Tor Vergata‘, Roma 00133, Italy; e-mail: caporaso@abel.harvard.edu
JOE HARRIS
Affiliation:
Mathematics Department, Harvard University, Cambridge MA 02138, USA; e-mail: harris@abel.harvard.edu
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Abstract

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A new method to approach enumerative questions about rational curves on algebraic varieties is described. The idea is to reduce the counting problems to computations on the Néron-Severi group of a ruled surface. Applications include a short proof of Kontsevich‘s formula for plane curves and the solution of the analogous problem for the Hirzebruch surface F$_3$.

Keywords

rational surfaceruled surfacerational curveenumerative geometrySeveri variety.
Type
Research Article
Information
Compositio Mathematica , Volume 113 , Issue 2 , September 1998 , pp. 209 - 237
Copyright
© 1998 Kluwer Academic Publishers