Open Access
October 2015 Hyperelliptic surfaces with $K^{2} < 4\chi - 6$
Carlos Rito, María Martí Sánchez
Osaka J. Math. 52(4): 929-947 (October 2015).


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.


Download Citation

Carlos Rito. María Martí Sánchez. "Hyperelliptic surfaces with $K^{2} < 4\chi - 6$." Osaka J. Math. 52 (4) 929 - 947, October 2015.


Published: October 2015
First available in Project Euclid: 18 November 2015

zbMATH: 1343.14036
MathSciNet: MR3426622

Primary: 14J29

Rights: Copyright © 2015 Osaka University and Osaka City University, Departments of Mathematics

Vol.52 • No. 4 • October 2015
Back to Top