Automorphisms of surfaces over fields of positive characteristic

We study automorphism and birational automorphism groups of varieties over fields of positive characteristic from the point of view of Jordan and p-Jordan property. In particular, we show that the Cremona group of rank 2 over a field of characteristic p > 0 is p-Jordan, and the birational automorphism group of an arbitrary geometrically irreducible algebraic surface is nilpotently p-Jordan of class at most 2. Also, we show that the automorphism group of a smooth geometrically irreducible projective variety of nonnegative Kodaira dimension is Jordan in the usual sense.