Actions of ${\mu }_p$ on canonically polarized surfaces in characteristic p>0

This paper studies the existence of non trivial ${\mu }_p$ actions on a canonically polarized surface X defined over an algebraically closed field of characteristic p>0. In particular, an explicit function $f(K_X^2)$ is obtained such that if $p>f(K_X^2)$, then there does not exist a non trivial ${\mu }_p$-action on X. This implies that the connected component of the automorphism scheme of X containing the identity is either smooth or is obtained by successive extensions by ${\alpha }_p$.