Abstract: A complex variety is called rationally connected if any two very general points can be connected by a rational curve.  In this talk, we will discuss subvarieties of the degenerations of rationally connected varieties and the relation of this problem with Ax conjecture which predicts that every perfect PAC field is C_1.  We will start with the proof of Ax conjecture by Jason Starr when the perfect PAC field contains an algebraically closed field.  Next we will talk about how we can generalize Jason's proof when the perfect field does not contain an algebraically closed field.  This approach produces a geometrically irreducible subvariety on the degeneration.  And we will show that the dimension of this subvariety has a lower bound by the Fano index of the generic fiber.