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arXiv · 1105.3102

Families of Log Canonically Polarized Varieties

Ariana Dundon
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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 $¶^1$, i.e. families $g:(Y,D)\to ¶^1$ where $D$ is an effective snc divisor and the sheaf $ω_{Y/¶^1}(D)$ 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.

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Ariana Dundon. 2011-05-16. Families of Log Canonically Polarized Varieties. https://arxiv.org/abs/1105.3102

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