Automorphisms of smooth canonically polarized surfaces in positive characteristic

This paper investigates the geometry of a smooth canonically polarized surface X defined over an algebraically closed field of characteristic p>0 in the case when the automorphism scheme of X is not smooth. In particular, it is shown that a smooth canonically polarized surface X with 1≤KX2≤2 and non-smooth automorphism scheme tends to be uniruled and simply connected and is the purely inseparable quotient of a ruled or rational surface by a rational vector field. Moreover, restrictions on certain numerical invariants of X are obtained in order for Aut(X) to be smooth.