Smooth hypersurfaces in abelian varieties over arithmetic rings

TL;DR

This paper proves the finiteness of smooth hypersurfaces representing an ample line bundle in abelian schemes over certain arithmetic rings, using moduli stack techniques and level structures.

Contribution

It establishes a finiteness result for smooth hypersurfaces in abelian varieties over arithmetic rings, connecting moduli stacks with level structures.

Findings

Finiteness of smooth hypersurfaces in abelian schemes over arithmetic rings.

Use of moduli stacks and level structures to interpolate finiteness results.

Application to hypersurfaces representing ample line bundles.

Abstract

Let $A$ be an abelian scheme of dimension at least four over a $\mathbb{Z}$-finitely generated integral domain $R$ of characteristic zero, and let $L$ be an ample line bundle on $A$. We prove that the set of smooth hypersurfaces $D$ in $A$ representing $L$ is finite by showing that the moduli stack of such hypersurfaces has only finitely many $R$-points. We accomplish this by using level structures to interpolate finiteness results between this moduli stack and the stack of canonically polarized varieties.