Actions of $\mu_p$ on canonically polarized surfaces in characteristic p>0

TL;DR

This paper investigates the conditions under which non-trivial bc_pb actions exist on canonically polarized surfaces in characteristic p>0, establishing bounds related to the surface's invariants and describing the automorphism scheme structure.

Contribution

It provides an explicit function bounding p for the existence of bc_pb-actions and characterizes the automorphism scheme of such surfaces in positive characteristic.

Findings

Non-trivial bc_pb-actions do not exist for p > f(K_X^2).

The automorphism scheme component is either smooth or built from extensions by b1_p.

An explicit function f(K_X^2) bounds the characteristic p for bc_pb-action existence.

Abstract

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 $\alpha_p$.