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

TL;DR

This paper investigates the properties of minimal surfaces of general type with hyperelliptic fibrations, establishing bounds on the genus and classifying branch loci based on the surface invariants.

Contribution

It proves genus bounds for hyperelliptic fibrations under certain numerical conditions and classifies branch loci for surfaces with specific invariants.

Findings

Genus g is bounded when K^2 < 4χ - 6.

Classification of branch curves for K^2 < 3χ - 6.

Maximum χ for given g > 4 and K^2 - 3χ < -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 $S\rightarrow S/i,$ 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.