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Parameter spaces for curves on surfaces and enumeration of rational curves

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 2, Roma 00133, Italy. caporaso@abel.harvard.edu
JOE HARRIS
Affiliation:
Mathematics department, Harvard University, Cambridge MA 02138, USA
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Abstract

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Let S be a smooth, minimal rational surface. The geometry of the Severi variety parametrising irreducible, rational curves in a given linear system on S is studied. The results obtained are applied to enumerative geometry, in combination with ideas from Quantum Cohomology. Formulas enumerating rational curves are found, some of which generalised Kontsevich‘s formula for plane curves.

Keywords

rational surfacerational curveenumerative geometrySeveri varietycross-ratio method.
Type
Research Article
Information
Compositio Mathematica , Volume 113 , Issue 2 , September 1998 , pp. 155 - 208
Copyright
© 1998 Kluwer Academic Publishers