On birationally trivial families and adjoint quadrics

Abstract

Copyright © 2022 The Author(s).Let π: X → B be a family whose general fibre Xb is a (d1, ⋯, da)-polarization on a general abelian variety, where 1 ≤ di ≤ 2, i = 1, ⋯, a and a ≥ 4. We show that the fibres are in the same birational class if all the (m, 0)-forms on Xb are liftable to (m, 0)-forms on X, where m = 1 and m = a - 1. Actually, we show a general criteria to establish whether the fibres of certain families belong to the same birational class.