Families of Log Canonically Polarized Varieties

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.