Pluricanonical maps of stable log surfaces

TL;DR

This paper extends classical results on pluricanonical maps to stable log surfaces, establishing bounds for base-point-freeness and very ampleness of multiples of the log canonical divisor.

Contribution

It generalizes Kodaira and Bombieri's results to stable log surfaces, providing explicit bounds for pluricanonical maps in this broader setting.

Findings

$4I(K_X+ riangle)$ is base-point-free

$8I(K_X+ riangle)$ is very ample

Bounds improve under specific singularity conditions

Abstract

Stable surfaces and their log analogues are the type of varieties naturally occuring as boundary points in moduli spaces. We extend classical results of Kodaira and Bombieri to this more general setting: if $(X,\Delta)$ is a stable log surface with reduced boundary (possibly empty) and $I$ is its global index, then $4I(K_X+\Delta)$ is base-point-free and $8I(K_X+\Delta)$ is very ample. These bounds can be improved under further assumptions on the singularities or invariants, for example, $5(K_X+\Delta)$ is very ample if $(X,\Delta)$ has semi-canonical singularities.