Model theory and geometry of fields with automorphism


I will review some of the model-theoretic geometry of difference varieties, and some open problems. 
A difference variety is defined by polynomial equations with an additional operator $\si$ interpreted as a field automorphism. 
For each prime power $q=p^m$ there is a functor $M_q$ from difference varieties (or schemes) to ordinary varieties, by specializing to characteristic $p$ and interpreting $\si$ as the Frobenius automorphism $x \mapsto x^q$.    Many constructions of algebraic geometry can be generalized so as to be compatible with these maps; notably, intersection theory on smooth projective varieties.   Zero-dimensional difference varieties (say over $\Qq$)
(whose jet space is a finite-dimensional pro-algebraic variety),  are at the heart of the theory;  in particular the intersection of two difference subvarieties of complementary dimension is not a number but a zero-dimensional difference variety, up to a certain model-theoretically defined equivalence.         Here the model theory of valued difference fields intervenes; specifically the notion of analyzability.    Zero-dimensional difference varieties generalize varieties over (pseudo)-finite fields, but unlike the latter they can be moved in families.  They appear to be closely connected to motives; I will formulate some questions and show a first instance of this conjectural connection.