Timeline for A Nodal curve embedded in a smooth variety, is always regularly embedded?
Current License: CC BY-SA 3.0
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| Oct 1, 2015 at 15:42 | history | edited | free-object | CC BY-SA 3.0 |
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| Oct 1, 2015 at 9:55 | comment | added | abx | In other words: locally around $p$ (for the classical or étale topology), the embedding $C\hookrightarrow Y$ is isomorphic to $C \hookrightarrow T_p(C)\hookrightarrow T_p(Y)$, which is regular because it is the composition of two regular embeddings. | |
| Oct 1, 2015 at 8:18 | comment | added | t3suji | I think so, it is a local complete intersection, like Allen Knutson says. | |
| Oct 1, 2015 at 7:17 | comment | added | free-object | t3suji//thanks, then you think that it is true that $C$ is embeded regularly in this situation?? | |
| Oct 1, 2015 at 7:06 | comment | added | t3suji | No, it's not possible: look at the generators of the ideal of C, and choose among those the elements whose differentials at p are linearly independent. They will cut out a subvariety that is smooth at p, and its tangent space is the same as C, which makes it a surface. | |
| Oct 1, 2015 at 5:55 | comment | added | free-object | If $C_1$ and $C_2$ meets transversally at $p$, is it possible that there is no smooth surface containing $C$ in a formal neighborhood?? | |
| Oct 1, 2015 at 4:47 | comment | added | Allen Knutson | This is just a question for the formal neighborhood of the node, right? And then it's enough to say that $C$ is a local complete intersection? Is it true that $C_1 \cup C_2$ lies in a smooth surface inside this formal neighborhood in $Y$? | |
| Oct 1, 2015 at 3:24 | history | asked | free-object | CC BY-SA 3.0 |