Families of reducible algebraic varieties (Q1387425)

Let \(f: A \rightarrow S\) be a non-constant family of abelian surfaces over a curve \(S\). Assume that the Picard number of the general fiber is 3 and that \(S\) has a non-constant morphism onto a projective line of degree \(n\). The set of the Néron-Severi lattices of the general fibers of all such families is finite and its cardinality is bounded by a constant depending only on \(n\). The same result was proved by the author for abelian surfaces with Picard number 4 defined over an algebraic number field of bounded degree over \(\mathbb{Q}\) [\textit{I. R. Shafarevich}, in: Algebra and Analysis, Proc. Int. Centrum, Chebotarev Conf., Kazan 1994, 103-108 (1996)].