Timeline for Maps to projective space determined by a line bundle
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| S Dec 30, 2013 at 9:13 | history | suggested | KotelKanim | CC BY-SA 3.0 |
texing some math stuff
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| Dec 30, 2013 at 9:00 | review | Suggested edits | |||
| S Dec 30, 2013 at 9:13 | |||||
| Jul 18, 2011 at 19:56 | comment | added | Martin Brandenburg | I don't get the point. The general definition of projective space in the functorial approach to algebraic geometry (see e.g. EGA I) just says that maps into projective space are given by invertible sheaves + global generators. The description of the points follows from it (plug in fields). If you pick a universal element, i.e. the serre twist $\mathcal{O}(1)$ with its chosen generators, then the bijection is on field-valued points exactly what is described above, but it should not be confused with a complete definition of the map into projective space. | |
| Jul 18, 2011 at 19:20 | answer | added | Anton Geraschenko | timeline score: 12 | |
| Nov 18, 2010 at 0:17 | answer | added | roy smith | timeline score: 23 | |
| Nov 8, 2009 at 15:30 | comment | added | Greg Muller | sigh I'm not saying that; what I was trying imply (crudely) is that the construction seemed to be very 'variety-theoretic', and not 'scheme-theoretic', in that you are defining a map on closed points and calling it a day. I know this construction works in scheme-y generality, but to even think about it over things like finite fields, there are better ways to think about the construction. I was certainly not trying to impune arithmetic algebraic geometry. | |
| Nov 8, 2009 at 8:25 | comment | added | Kevin Buzzard | I'm still confused by what you say :-) At the minute you seem to be saying "I don't believe in algebraic geometry over finite fields"... | |
| Nov 8, 2009 at 3:11 | comment | added | Greg Muller | Oh, just to be a bit clear, when I mentioned generalizing things to other fields than C, it wasn't because there's any dependence on being algebraically closed. I meant that the construction was geometric, and so it seemed somewhat unnatural to do it over a finite field. | |
| Nov 8, 2009 at 0:02 | comment | added | Andrew Critch | Great question. Re: "This constructions is remarkably ad hoc", see Anton's answer, which I think makes it easiest to see how non- ad hoc these morphisms really are! | |
| Nov 7, 2009 at 23:52 | answer | added | Anton Geraschenko | timeline score: 20 | |
| Nov 7, 2009 at 23:27 | answer | added | Ben Webster♦ | timeline score: 18 | |
| Nov 7, 2009 at 23:09 | answer | added | Ilya Nikokoshev | timeline score: 3 | |
| Nov 7, 2009 at 22:59 | comment | added | Kevin Buzzard | I think it's usually called the "complete linear system" determined by the line bundle. It works over all fields and even Z. If you want to know what a line bundle is in this context, read Hartshorne (or any graduate-level alg geom book), which gives an intrinsically algebraic way to construct the map. | |
| Nov 7, 2009 at 22:51 | history | asked | Greg Muller | CC BY-SA 2.5 |