Timeline for Function Fields of Real Varieties
Current License: CC BY-SA 2.5
10 events
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| Dec 2, 2010 at 10:29 | comment | added | Pete L. Clark | @James: Yes, absolutely. In retrospect, I could have written this in a less confusing way. I find this quirk of real algebraic geometry amusing, but it's certainly not necessary to mention it in this context: it's pretty clear that the result in question extends to non-affine varieties. | |
| Dec 2, 2010 at 9:04 | comment | added | JBorger | Thanks. So, by "real algebraic variety" you don't mean an algebraic variety (in any of the standard scheme-theoretic senses) over the field of real numbers. Dangerous terminology! | |
| Nov 30, 2010 at 13:15 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
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| Nov 30, 2010 at 13:04 | comment | added | Pete L. Clark | For instance $\mathbb{P}^1$ is, as a real algebraic variety, isomorphic to $S^1= \operatorname{Spec}(\mathbb{R}[x,y]/(x^2+y^2−1))$. | |
| Nov 30, 2010 at 13:00 | comment | added | Pete L. Clark | See Chapter 3 of Bochnak, Coste and Roy for more details, but briefly: one has the notion of a real algebraic variety, a locally ringed space locally isomorphic to the set of $\mathbb{R}$-points of an affine $\mathbb{R}$-variety with the analytic topology. Of course projective space $\mathbb{P}^n$ is a real algebraic variety. But as a real algebraic variety, it is affine. | |
| Nov 30, 2010 at 12:05 | comment | added | JBorger | Dear Pete, can you clarify what you mean by projective varieties over $\mathbf{R}$ being affine? The projective line over $\mathbf{R}$ is a counter-example if you take the usual, scheme-theoretic interpretations of these terms. | |
| Nov 30, 2010 at 11:44 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
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| Oct 28, 2010 at 0:15 | vote | accept | unknown | ||
| Oct 27, 2010 at 18:59 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
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| Oct 27, 2010 at 14:21 | history | answered | Pete L. Clark | CC BY-SA 2.5 |