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 like rational surfaces do, though it is not rational. The square of the first Chern class (and moreover the canonical class is ample).

Hodge diamond
1
00
090
00
1
gollark: Also, they can't emit IR and cook me, *or* emit (much) RF and probably somewhat break electronic stuff.
gollark: Obviously we need monitors which can properly represent laser videos, by blinding oyu.
gollark: To be able to emit ionizing radiation, yes.
gollark: Most monitors can't even generate a lot of *visible* spectrum colors, even. There are a bunch of color space diagrams of this on the internet, except they're not a very good way to show it because, unsurprisingly, the cyan-ish bit they can't display well just looks like identical cyan.
gollark: That would just allow per-*column* control, unless you scan them left and right really fast.

See also

References
  • Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004), Compact Complex Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 4, Springer-Verlag, Berlin, ISBN 978-3-540-00832-3, MR 2030225
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.