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I'm aware that the complex projective line is a sphere, the Riemann sphere, and the dual projective line is a cylinder; but I can't find anything mentioning what shape the split-complex projective line would be. I probably could figure it out for myself, of course, but I'm surprised that it doesn't come up in searching.

Seth Schmidt-O'Hainle
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  • It's probably not that straightforward, since it's not a field. What definition of projective line would you propose? – Berci Apr 16 '20 at 16:20
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    Duals aren't a field either, but they have a projective line. The definition is rather standard, isn't it? Ordered pairs (x,y) together with an equivalence relation ~ where for all invertible elements n you have (x,y) ~ (xn,yn). – Seth Schmidt-O'Hainle Apr 17 '20 at 01:51
  • It's a hyperboloid of one sheet, see for example the third paragraph here. More generally, if I'm not mistaken the geometry of the projective line over the ring $\mathbb{R}[j]/(j^2-t)$ is the surface of revolution of the curve $|x+yj|^2=x^2-ty^2=1$ around the $y$-axis. – pregunton Jun 14 '20 at 14:21
  • Thanks! I somehow never found that. If you could write that up as a full answer (and if possible include references to why that thing about surfaces of revolution is true) that would be great! – Seth Schmidt-O'Hainle Jun 14 '20 at 14:38

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