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.14E07, 14J50 1. Introduction 2748 2. Preliminaries 2752 3. Automorphism groups 2759 4. Group of connected components 2762 5. Minimal models 2764 6. Abelian varieties 2768 7. Varieties of nonnegative Kodaira dimension 2771 8. Jordan property for curves 2774 9. Cremona group 2776 10.Jordan property for surfaces 2778 11