Stable maps and branch divisors

1999/05/18 by Barbara Fantechi, B. Fantechi, Fantechi, B. +3 · 5 citations
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.AG

paper · pdf · doi:10.48550/arxiv.math/9905104

21 pages, LaTex2e

arxiv created 1999/05/18 · openalex publication_date 1999/05/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 P¹ for all genera and degrees in terms of Hodge integrals.

Cited by

• Logarithmic series and Hodge integrals in the tautological ring (with an appendix by D. Zagier)
• The Gromov-Witten potential of a point, Hurwitz numbers, and Hodge integrals
• Admissible covers and stable maps
• Hodge integrals and Hurwitz numbers via virtual localization
• Gromov-Witten invariants of Fano threefolds of genera 6 and 8

Related

• The Moduli Space of Stables Maps with Divisible Ramification
• Towards Mori's program for the moduli space of stable maps
• The Geometry of Stable Quotients in Genus One
• Branch Stabilisation for the Components of Hurwitz Moduli Spaces of\n Galois Covers
• Stable maps and Hurwitz schemes in mixed characteristic