Bound of automorphisms of fibred surfaces in positive characteristic

Abstract

Let $f:S\to B$ be a fibred surface over an algebraically closed field k of characteristic $p\ge 5$, where S is a minimal smooth projective surface of general type and B is a smooth projective curve of genus $b\ge 2$. We prove that the group of fibration-preserving automorphisms of f has order at most $15658{\left(K_S^2\right)}^4$. Furthermore, we provide an example to show that the exponent 4 of the polynomial bound is sharp.