Uniqueness of minimal morphisms of logarithmic schemes

TL;DR

This paper establishes conditions ensuring the finiteness and projectivity of moduli spaces of logarithmic scheme morphisms, advancing the understanding of their geometric structure and properties.

Contribution

It provides a new sufficient condition for the quasifiniteness of the moduli space of morphisms between logarithmic schemes, with implications for stable maps.

Findings

Moduli space of morphisms is quasifinite under certain conditions.

Finite over the moduli space of stable maps.

Existence of a projective coarse moduli space when the target is projective.

Abstract

We give a sufficient condition under which the moduli space of morphisms between logarithmic schemes is quasifinite under the moduli space of morphisms between the underlying schemes. This implies that the moduli space of stable maps from logarithmic curves to a target logarithmic scheme is finite over the moduli space of stable maps, and therefore that it has a projective coarse moduli space when the target is projective.