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 ${\mathbb {P}^1}$ , i.e. families ${g:(Y, D) \to \mathbb {P}^1}$ where D is an effective snc divisor and the sheaf ${\omega_{Y/\mathbb {P}^1}(D)}$ 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, ${\mathcal {O}_Y(D)}$ 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.