Smooth hypersurfaces in abelian varieties over arithmetic rings
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Let A be an abelian scheme of dimension at least four over a 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.
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Forum of Mathematics, Sigma, 10, Cambridge University Press, Cambridge, 2022, https://doi.org/10.1017/fms.2022.87