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Shafarevich's conjecture for families of hypersurfaces over function fields (2410.06387v1)
Published 8 Oct 2024 in math.AG
Abstract: Given a smooth quasi-projective complex algebraic variety $\mathcal{S}$, we prove that there are only finitely many Hodge-generic non-isotrivial families of smooth projective hypersurfaces over $\mathcal{S}$ of degree $d$ in $\mathbb{P}{\mathbb C}{n+1}$. We prove that the finiteness is uniform in $\mathcal{S}$ and we give examples where the result is sharp. We also prove similar results for certain complete intersections in $\mathbb{P}{\mathbb C}{n+1}$ of higher codimension and more generally for algebraic varieties whose moduli space admits a period map that satisfies the infinitesimal Torelli theorem.