Timeline for Set of Curves Passing through a smooth point of a Variety is Zariski-Dense
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 22, 2012 at 7:46 | comment | added | Angelo | I guess I was reading "smooth" as "smooth at the rational point". | |
| Aug 22, 2012 at 5:35 | comment | added | Karl Schwede | Oh that's true. If curves need not be closed then certainly things are fine. | |
| Aug 21, 2012 at 21:55 | comment | added | M P | Do you want the curves to be closed in your variety? If so, then this will be difficult: for instance, if the variety itself is a singular curve, then you won't even find a single smooth curve! Otherwise, I agree with Angelo. | |
| Aug 21, 2012 at 21:14 | comment | added | Karl Schwede | Dear Angelo, I'm confused slightly, do you need normality? Suppose that $X$ is a non-normal projective variety whose normalization is for example $\mathbb{P}^2_{\mathbb{C}}$. Suppose that the non-normal locus is a curve (I can do this with a pinch point). Then every hyperplane section will have a singularity, right? Of course, we can find tons of hyperplanes that are smooth in a neighborhood of the given point. | |
| Aug 21, 2012 at 19:18 | comment | added | Angelo | You can reduce to the affine case; then take hyperplane sections. | |
| Aug 21, 2012 at 18:54 | comment | added | Charles Staats | A note: The statement, as written, seems to have a lot to do with rationality over $K$. If this is your intent, then the title is a bit deceptive, since it suggests you might, for instance, care primarily about what happens when $K$ is algebraically closed. $$ $$ It might also be helpful to state what paper you are looking at, and where in the paper this statement appears. | |
| Aug 21, 2012 at 17:53 | history | asked | Nikesh | CC BY-SA 3.0 |