A Conjecture of Ax and Degenerations of Fano Varieties

A field k is called C1 if every homogeneous form f(x0, . . . , xn) ∈ k[x0, . . . , xn] of degree ≤ n has a nontrivial zero. Examples of C1 fields are finite fields (Chevalley) and function fields of curves over an algebraically closed field (Tsen). A field is called PAC (pseudo algebraically closed) if every geometrically integral k-variety has a k-point. An k-variety X is called geometrically integral (or absolutely irreducible) if it is still irreducible and reduced as a variety over the algebraic closure k̄.) These fields were introduced in [Ax68]; see [FJ05] for an exhaustive and up to date treatment. The aim of this paper is to prove in characteristic 0 a conjecture of Ax, posed in [Ax68, Problem 3].