Unlikely, likely and impossible intersections without algebraic groups

Abstract

We formulate function field analogues for the Zilber-Pink Conjecture and for the Bounded Height Conjecture. The ‘‘special’’ varieties in our formulation are varieties defined over the constant field. We prove our function field Zilber-Pink Conjecture for all subvarieties, and we prove our function field Bounded Height Conjecture for a certain class of curves. We explain that for our problems the algebraic groups are no longer ‘‘special’’; instead the relevant notion is the transcendence degree over the constant field of the field of definition for our varieties.