Automorphisms of smooth canonically polarized surfaces in characteristic 2

Abstract

This paper investigates the structure of the automorphism scheme of a smooth canonically polarized surface $X$ defined over an algebraically closed field of characteristic 2. In particular it is investigated when Aut(X) is not smooth. This is a situation that appears only in positive characteristic and it is closely related to the structure of the moduli stack of canonically polarized surfaces. Restrictions on certain numerical invariants of X are obtained in order for Aut(X) to be smooth or not and information is provided about the structure of the component of Aut(X) containing the identity. In particular, it is shown that if X is a smooth canonically polarized surface with 1\leq K_X^2 \leq 2 with non smooth automorphism scheme, then X is uniruled. Moreover, if K_X^2=1, then X is simply connected, unirational and p_g(X) \leq 1. Moreover, X is the purely inseparable quotient of a rational surface by a rational vector field.