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Godeaux surface
One of the surfaces of general type introduced by Lucien Godeaux in 1931
One of the surfaces of general type introduced by Lucien Godeaux in 1931
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).
| 1 | 0 | 0 | 0 | 9 | 0 | 0 | 0 | 1
References
This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.
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