Shafarevich's conjecture for families of hypersurfaces over function fields

2024/10/08 by Philip Engel, Engel, Philip, Alice Lin +3
Mathematics · Computer Science · #Meromorphic and Entire Functions #Algebraic Geometry and Number Theory #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2410.06387

Abstract

Given a smooth quasi-projective complex algebraic variety S, we prove that there are only finitely many Hodge-generic non-isotrivial families of smooth projective hypersurfaces over S of degree d in ℙ_{\mathbb C}ⁿ⁺¹. We prove that the finiteness is uniform in S and we give examples where the result is sharp. We also prove similar results for certain complete intersections in 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.

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