Compositio Mathematica



Parameter spaces for curves on surfaces and enumeration of rational curves


LUCIA CAPORASO a1 and JOE HARRIS a2
a1 Mathematics department, Harvard University, Cambridge MA 02138, USA and Dipartimento di Matematica, Universitá di Roma 2, Roma 00133, Italy. caporaso@abel.harvard.edu
a2 Mathematics department, Harvard University, Cambridge MA 02138, USA

Article author query
caporaso l   [Google Scholar] 
harris j   [Google Scholar] 
 

Abstract

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.


Key Words: rational surface; rational curve; enumerative geometry; Severi variety; cross-ratio method..