On uniformly effective birationality and the Shafarevich Conjecture over curves

TL;DR

This paper establishes an effective upper bound on the number of deformation types of certain families of canonically polarized manifolds over a curve, depending only on specific invariants, by studying effective birationality.

Contribution

It introduces a new approach to bounding deformation types using effective birationality for families of canonically polarized manifolds.

Findings

Bound depends only on genus, subset size, dimension, and volume.

Provides explicit effective bounds for deformation types.

Advances understanding of degeneracy loci in families of polarized manifolds.

Abstract

Let $B$ be a smooth projective curve of genus $g$, and $S \subset B$ be a finite subset of cardinality $s$. We give an effective upper bound on the number of deformation types of admissible families of canonically polarized manifolds of dimension $n$ with canonical volume $v$ over $B$ with prescribed degeneracy locus $S$. The effective bound only depends on the invariants $g, s, n$ and $v$. The key new ingredient which allows for this kind of result is a careful study of effective birationality for families of canonically polarized manifolds.