On degenerations of $\mathbb{Z}/2$-Godeaux surfaces

Eduardo Dias; Carlos Rito; Giancarlo Urzúa

doi:10.4171/rmi/1376

JournalsrmiVol. 38, No. 5pp. 1399–1423

On degenerations of $Z /2$ -Godeaux surfaces

Eduardo Dias
Faculdade de Ciências da Universidade do Porto, Portugal
Carlos Rito
Universidade de Trás-os-Montes e Alto Douro, Vila Real, Portugal
Giancarlo Urzúa
Pontificia Universidad Católica de Chile, Santiago de Chile, Chile

$On degenerations of $\mathbb{Z}/2$-Godeaux surfaces cover$

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Abstract

We compute equations for the Coughlan’s family of Godeaux surfaces with torsion $Z /2$ , which we call $Z /2$ -Godeaux surfaces, and we show that it is (at most) 7 dimensional. We classify all non-rational KSBA degenerations $W$ of $Z /2$ -Godeaux surfaces with one Wahl singularity, showing that $W$ is birational to particular either Enriques surfaces, or $D_{2, n}$ elliptic surfaces, with $n = 3, 4$ or $6$ . We present examples for all possibilities in the first case, and for $n = 3, 4$ in the second.

Cite this article

Eduardo Dias, Carlos Rito, Giancarlo Urzúa, On degenerations of $Z /2$ -Godeaux surfaces. Rev. Mat. Iberoam. 38 (2022), no. 5, pp. 1399–1423

DOI 10.4171/RMI/1376