Extending hyperelliptic K3 surfaces, and Godeaux surfaces with $\pi_1=\mathbb{Z}/2$
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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 $\pi_1=\ZZ/2$.
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 $\pi_1=\ZZ/2$.
참고문헌 (12)
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