Hyperelliptic surfaces with $K^{2} < 4\chi - 6$

Abstract

Let $S$ be a smooth minimal surface of general type with a (rational) pencil of hyperelliptic curves of minimal genus $g$. We prove that if $K_S^2 4$ and $K_S^2-3\chi(\mathcal O_S)<-6,$ we compute the maximum value for $\chi(\mathcal O_S).$ This list of possibilities is sharp.