Stable maps and branch divisors

TL;DR

This paper develops a new method to construct branch divisors for certain morphisms with singular domains, enabling the computation of Hurwitz numbers via stable maps and Hodge integrals.

Contribution

It generalizes Mumford's divisor construction to complexes and applies it to stable maps, providing a canonical branch morphism for enumerative geometry.

Findings

Constructed a natural branch divisor for morphisms with lci singularities.

Established a canonical branch morphism from stable maps to symmetric products.

Enabled computation of Hurwitz numbers for all genera and degrees.

Abstract

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 contruction 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 P^1 for all genera and degrees in terms of Hodge integrals.