Pink’s conjecture for abelian schemes

Establish that, for an abelian scheme A over a normal quasi-projective variety S and a subvariety X of dimension d dominating S with geometrically irreducible generic fiber, if X is not contained in any strict group subscheme of A, then X intersected with the union A_(<g-d) of fiberwise algebraic subgroups of relative dimension less than g-d is not Zariski dense in X.

Background

The paper studies the abelian-scheme formulation of Pink’s Zilber–Pink conjecture. For an abelian scheme A/S of relative dimension g, the locus A_(<t) is the union of algebraic subgroups of dimension less than t in the geometric fibers. Pink’s prediction asserts nondensity of the intersection with A_(<g-d) when X has dimension d, dominates the base, has geometrically irreducible generic fiber, and is not contained in a strict group subscheme.

The paper proves a finiteness theorem for intersections with flat group subschemes outside an explicitly defined degeneracy locus, thereby establishing substantial cases of the prediction, but it does not resolve the conjecture for arbitrary algebraic subgroups in general.