The logarithmic Picard group and its tropicalization

TL;DR

This paper develops the theory of logarithmic and tropical Picard groups for logarithmic curves, showing their relationships and properties, including the properness of the logarithmic Jacobian and the representability of the Picard group via modifications.

Contribution

It constructs the logarithmic and tropical Picard groups, relates them through quotient structures, and analyzes their geometric properties and representability.

Findings

Logarithmic Jacobian is a proper family over the moduli space.

Tropical Picard group is obtained as a quotient of the logarithmic Picard group.

Logarithmic Picard group admits modifications representable by logarithmic schemes.

Abstract

We construct the logarithmic and tropical Picard groups of a family of logarithmic curves and realize the latter as the quotient of the former by the algebraic Jacobian. We show that the logarithmic Jacobian is a proper family of logarithmic abelian varieties over the moduli space of Deligne-Mumford stable curves, but does not possess an underlying algebraic stack. However, the logarithmic Picard group does have logarithmic modifications that are representable by logarithmic schemes, all of which are obtained by pullback from subdivisions of the tropical Picard group.