Timeline for Zariski closures of one parameter additive maps in positive characteristic
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 30, 2011 at 17:28 | vote | accept | Confused | ||
| Oct 29, 2011 at 22:38 | comment | added | Confused | Also, your original "answer" comment made me wonder, are the only types of singularities for an algebraic curve "cusps" (is this $d\Theta =0$?) and "double points"? I guess the answer comment is checking for these "double points"? Thanks again for the interesting (for me anyways!) discussion. | |
| Oct 29, 2011 at 22:32 | comment | added | Confused | I need to think about an exact statement of the inverse function theorem in this setting, probably it's obvious, but would it imply that these algebraically defined tangent spaces are isomorphic? | |
| Oct 29, 2011 at 22:31 | comment | added | Confused | This last comment is exactly the issue that I'm worried about, that the $d\Theta$ may not be surjective. Like you say, certainly the tangent space at the identity of the image contains a vector $(\Theta_1'(0), \ldots, \Theta_{\delta}'(0))$, however how does one know that that is everything? As far as I know, this is related to the separability of $\Theta$, which seems to me to be impossible to check directly. | |
| Oct 29, 2011 at 22:01 | comment | added | Felipe Voloch | Or better, the tangent space of $\Phi({\mathbb A}^1)$ at $\Phi(a)$ is the image of the tangent space of ${\mathbb A}^1$ at $a$ by $d\Phi_a$ or something. Calculus! | |
| Oct 29, 2011 at 21:50 | comment | added | Felipe Voloch | Just use the implicit function theorem. | |
| Oct 29, 2011 at 20:34 | comment | added | Confused | Thanks again for your help. What characterization of tangent space are you using? My entire problem seems to revolve around knowing very little about the coordinate ring of the image variety, and the tangent space characterizations I know require knowledge of the ideal of functions which vanish on the image in some form or another. (which I don't really have) I'm still unsure how to rule out that the tangent space at say the identity doesn't have dimension greater than 1. Is there a source that you can recommend in addition to your comment? | |
| Oct 29, 2011 at 17:17 | history | answered | Felipe Voloch | CC BY-SA 3.0 |