Divisors and curves on logarithmic mapping spaces

TL;DR

This paper computes the rational class and Picard groups of genus zero logarithmic mapping spaces with a projective target, revealing explicit bases and relations derived from tropical and WDVV structures.

Contribution

It provides explicit bases for the class and Picard groups of these moduli spaces and introduces a technique for constructing test curves to analyze their topology.

Findings

Explicit basis for the class group with boundary divisors

Spanning set for the Picard group indexed by tropical functions

Complete boundary relations derived from WDVV relations

Abstract

We determine the rational class and Picard groups of the moduli space of stable logarithmic maps in genus zero, with target projective space relative a hyperplane. For the class group we exhibit an explicit basis consisting of boundary divisors. For the Picard group we exhibit a spanning set indexed by piecewise-linear functions on the tropicalisation. In both cases a complete set of boundary relations is obtained by pulling back the WDVV relations from the space of stable curves. Our proofs hinge on a controlled technique for manufacturing test curves in logarithmic mapping spaces, opening up the topology of these spaces to further study.