Families of Log Canonically Polarized Varieties

TL;DR

This paper investigates families of log canonically polarized varieties over the projective line, establishing conditions under which such families are either trivial or have at least three singular fibers, extending understanding of their moduli and singularities.

Contribution

It introduces a framework for families of log canonically polarized varieties over  and proves a lower bound on the number of singular fibers under mild hypotheses.

Findings

Families are either isotrivial or have at least 3 singular fibers.

Defines singular fibers for log canonically polarized families.

Establishes conditions for non-trivial families with few singular fibers.

Abstract

Determining the number of singular fibers in a family of varieties over a curve is a generalization of Shafarevich's Conjecture and has implications for the types of subvarieties that can appear in the corresponding moduli stack. We consider families of log canonically polarized varieties over $\P^1$, i.e. families $g:(Y,D)\to \P^1$ where $D$ is an effective snc divisor and the sheaf $\omega_{Y/\P^1}(D)$ is $g$-ample. After first defining what it means for fibers of such a family to be singular, we show that with the addition of certain mild hypotheses (the fibers have finite automorphism group, $\sO_Y(D)$ is semi-ample, and the components of $D$ must avoid the singular locus of the fibers and intersect the fibers transversely), such a family must either be isotrivial or contain at least 3 singular fibers.