Bounded height in pencils of finitely generated subgroups

In this article we prove a general bounded height result for specializations in finitely generated subgroups varying in families which complements and sharpens the toric Mordell-Lang theorem by replacing finiteness with emptiness, for the intersection of varieties and subgroups, all moving in a pencil, except for bounded height values of the parameters (and excluding identical relations). More precisely, an instance of the result is as follows. Consider the torus scheme G m/Cr over a curve C defined over Q, and let Γ be a subgroup scheme generated by finitely many sections (satisfying some necessary conditions). Further, let V be any subscheme. Then there is a bound for the height of the points P ∈ C (Q) such that, for some γ ∈ Γ which does not generically lie in V, γ (P) lies in the fiber VP. We further offer some direct Diophantine applications, to illustrate once again that the results implicitly contain information absent from the previous bounds in this context.