The torsion group of a certain numerical Godeaux surface

Abstract

We compute the torsion part of the Picard group of a numerical Godeaux surface Y which was constructed by Stagnaro [7] as a double cover of $\mathbb{P}^{2}$ branching along a curve of degree 10. We show that this surface is a classical Godeaux surface whose universal cover is the Fermat quintic in $\mathbb{P}^{3}$ (cf. [1, p.170]).