Extending hyperelliptic K3 surfaces, and Godeaux surfaces with $\pi_1=\mathbb{Z}/2$

Stephen Coughlan

doi:10.4134/JKMS.j150307

Journal of the
Korean Mathematical Society JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2016; 53(4): 869-893

Printed July 1, 2016

https://doi.org/10.4134/JKMS.j150307

Extending hyperelliptic K3 surfaces, and Godeaux surfaces with $\pi_1=\mathbb{Z}/2$

Stephen Coughlan

Leibniz Universit\"at Hannover

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Abstract

We construct the extension of a hyperelliptic K3 surface to a Fano $6$-fold with extraordinary properties in moduli. This leads us to a family of surfaces of general type with $p_g=1$, $q=0$, $K^2=2$ and hyperelliptic canonical curve, each of which is a weighted complete intersection inside a Fano $6$-fold. Finally, we use these hyperelliptic surfaces to determine an $8$-parameter family of Godeaux surfaces with $\pi_1=\ZZ/2$.

Keywords: surfaces of general type, Godeaux surfaces, Fano $6$-folds

MSC numbers: 14J10, 14J29, 14J40