  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. 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 a smooth canonically polarized surface X with 0< K_X^2 < 3 and non smooth
automorphism scheme tends to be uniruled and simply connected. Moreover, X is
the purely inseparable quotient of a ruled or rational surface by a rational
vector field.
