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Stable maps and branch divisors. (English) Zbl 1054.14033
Summary: 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.

##### MSC:
 14H10 Families, moduli of curves (algebraic) 14C20 Divisors, linear systems, invertible sheaves 14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
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