Shafarevich’s conjecture for families of hypersurfaces over function fields

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^{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^{n+1} of higher codimension and more generally for algebraic varieties whose moduli space admits a period map that satisfies the infinitesimal Torelli theorem.

This is joint work with Philip Engel and Salim Tayou.

See webpage for more details: https://people.math.harvard.edu/~ctm/sem/