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 $$q = 0, p_g = 0$$ like rational surfaces do, though it is not rational. The square of the first Chern class $$c_1^2 = 1$$ (and moreover the canonical class is ample).