Timeline for Path connectedness of varieties
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 21, 2013 at 22:44 | answer | added | Sean Lawton | timeline score: 7 | |
| Apr 26, 2011 at 0:06 | vote | accept | Brian | ||
| Apr 25, 2011 at 2:50 | answer | added | roy smith | timeline score: 46 | |
| Apr 24, 2011 at 22:29 | comment | added | Pete L. Clark | We have some more than adequate answers given in the comments. Would one of the commenters be willing to step up and actually answer the question in the formal MO sense? | |
| Apr 24, 2011 at 15:30 | comment | added | Brian | Dear Roy Smith: Thanks a lot for your explanation about using blowing up so that Bertini (the form given in Hartshorne) can be applied. | |
| Apr 24, 2011 at 15:29 | comment | added | Brian | Dear Karl Schwede: Thanks a lot for your answer. My question about the curve is indeed a very dumb one. | |
| Apr 24, 2011 at 15:24 | comment | added | Karl Schwede | Brian, J.C. Ottem is right. You can just use Bertini. To your question of whether every curve is the image of a non-singular one, the answer is yes, just take the normalization of the curve (see the section on curves in the first chapter of Hartshorne). I don't know what you mean by segment on a curve though. | |
| Apr 24, 2011 at 14:46 | comment | added | Brian | The version in Hartshorne requires $X$ has at most a finite number of singular points and that $X$ projective (or equivalently, projective with a finite number of points removed). Do you have a more general form in mind? Also, your answer leads to another question (probably a dumb one that I cannot think of): curves are parametrizable, i.e. any segment on a curve is an image of a non-singular curve? | |
| Apr 24, 2011 at 14:36 | comment | added | J.C. Ottem | If $X$ is quasi-projective and of dim $\ge 2$, you can use Bertini's theorem on a sufficiently general hyperplane section through P and Q. | |
| Apr 24, 2011 at 14:18 | history | asked | Brian | CC BY-SA 3.0 |