On uniformly effective birationality and the Shafarevich Conjecture over curves

Details On uniformly effective birationality and the Shafarevich Conjecture over curves On uniformly effective birationality and the Shafarevich Conjecture over curves

2011-05-17

arxiv.org/abs/1105.3439v1

Mathematics

Algebraic Geometry

22 pages

Scientific paper

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.

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