The Ax conjecture predicts that every PAC (pseudo-algracially closed) field is C1. This conjecture has been completely solved when the field has characteristic zero. Also, the conjecture is answered positively when the field contains an algebraically closed field. The arithmetic set up of Ax conjecture is closely related to the geometry of degenerations of Fano varieties in characteristic zero, and degenerations of separably rationally connected varieties in positive characteristics. This geometric set up predicts the existence of a geometrically irreducible subscheme on the degenerations. In this talk, we will give a brief review on known results about Ax conjecture and then talk about how to bound the dimensions of these geometrically irreducible subschemes on degenerations.