# Godeaux surface

In mathematics, a Godeaux surface is one of the surfaces of general type introduced by Lucien Godeaux in 1931. Other surfaces constructed in a similar way with the same Hodge numbers are also sometimes called Godeaux surfaces. Surfaces with the same Hodge numbers (such as Barlow surfaces) are called numerical Godeaux surfaces.

## Construction

The cyclic group of order 5 acts freely on the Fermat surface of points (w : x : y : z) in P3 satisfying w5 + x5 + y5 + z5 = 0 by mapping (w : x : y : z) to (w:ρx:ρ2y:ρ3z) where ρ is a fifth root of 1. The quotient by this action is the original Godeaux surface.

## Invariants

The fundamental group (of the original Godeaux surface) is cyclic of order 5. It has invariants ${\displaystyle q=0,p_{g}=0}$ like rational surfaces do, though it is not rational. The square of the first Chern class ${\displaystyle c_{1}^{2}=1}$ (and moreover the canonical class is ample).

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