Smooth hypersurfaces in abelian varieties over arithmetic rings. (English) Zbl 1499.14048
Summary: 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.
MSC:
| 14G99 | Arithmetic problems in algebraic geometry; Diophantine geometry |
| 14D23 | Stacks and moduli problems |
| 11G35 | Varieties over global fields |
| 14G05 | Rational points |
| 32Q45 | Hyperbolic and Kobayashi hyperbolic manifolds |
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