## Osaka Journal of Mathematics

### 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.

#### Article information

Source
Osaka J. Math., Volume 52, Number 4 (2015), 929-947.

Dates
First available in Project Euclid: 18 November 2015

https://projecteuclid.org/euclid.ojm/1447856026

Mathematical Reviews number (MathSciNet)
MR3426622

Zentralblatt MATH identifier
1343.14036

Subjects
Primary: 14J29: Surfaces of general type

#### Citation

Rito, Carlos; Sánchez, María Martí. Hyperelliptic surfaces with $K^{2} &lt; 4\chi - 6$. Osaka J. Math. 52 (2015), no. 4, 929--947. https://projecteuclid.org/euclid.ojm/1447856026

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