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Stable Maps and Branch Divisors

Published online by Cambridge University Press:  04 December 2007

B. Fantechi  and
R. Pandharipande
B. Fantechi
Affiliation:
Barbara Fantechi, Dipartimento di Matematica, Università di Trento, Via Sommarive 14, 38050 Povo, Italy. E-mail: fantechi@alpha.science.unitn.it
R. Pandharipande
Affiliation:
Rahul Pandharipande, Mathematics 253-37, Caltech, Pasadena, CA 91125, U.S.A. E-mail: rahulp@cco.caltech.edu
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Abstract

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We construct a natural branch divisor for equidimensional projective morphisms where the domain has lci singularities and the target is nonsingular. The method involves generalizing a divisor construction of Mumford from sheaves to complexes. The construction is valid in flat families. The generalized branch divisor of a stable map to a nonsingular curve X yields a canonical morphism from the space of stable maps to a symmetric product of X. This branch morphism (together with virtual localization) is used to compute the Hurwitz numbers of covers of the projective line for all genera and degrees in terms of Hodge integrals.

Keywords

Branch divisorHurwitz number
Type
Research Article
Information
Compositio Mathematica , Volume 130 , Issue 3 , February 2002 , pp. 345 - 364
Copyright
© 2002 Kluwer Academic Publishers