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 $K_{S}^{2} < 4\chi(\mathcal{O}_{S})-6$, then $g$ is bounded. The surface $S$ is determined by the branch locus of the covering $S \to S/i$, where $i$ is the hyperelliptic involution of $S$. For $K_{S}^{2} < 3\chi(\mathcal{O}_{S})-6$, we show how to determine the possibilities for this branch curve. As an application, given $g > 4$ 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.