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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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