  This paper investigates the geometry of smooth canonically polarized surfaces
defined over a field of positive characteristic which have a nontrivial global
vector field, and the implications that the existence of such surfaces has in
the moduli problem of canonically polarized surfaces.
  In particular, an explicit real valued function f(x) is obtained such that if
$X$ is a smooth canonically polarized surface defined over an algebraically
closed field of characteristic p>0 such that $K_X^2 <f(p)$, then $X$ is
unirational and the order of its algebraic fundamental group is at most two. As
a consequence of this result, large classes of canonically polarized surfaces
are identified whose moduli stack is Deligne-Mumford, a property that does not
hold in general in positive characteristic.
