On birationally trivial families and adjoint quadrics. Issue 3 (4th August 2022)

• Record Type: Journal Article
• Authors: Cesarano, Luca; Rizzi, Luca; Zucconi, Francesco
• Is Part Of: Proceedings of the Edinburgh Mathematical Society. Volume 65:Issue 3(2022)
• Page Start: 587
• Page End: 617
• Publication Date: 2022-08-04
• Subjects: extension class of a vector bundle -- holomorphic forms -- Albanese variety -- families of varieties -- infinitesimal invariant; 14J10 -- 14C34 -- 14D07 -- 14E99 -- 14J40; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0013091522000232
• Languages: English
• ISSNs: 0013-0915

Abstract

Abstract: Let $\pi \colon \mathcal {X}\to B$ be a family whose general fibre $X_b$ is a $(d_1, \, \ldots, \, d_a)$ -polarization on a general abelian variety, where $1\leq d_i\leq 2$, $i=1, \, \ldots, \, a$ and $a\geq 4$ . We show that the fibres are in the same birational class if all the $(m, \, 0)$ -forms on $X_b$ are liftable to $(m, \, 0)$ -forms on $\mathcal {X}$, 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.