Timeline for Questions on theorem in Deligne-Mumford's '69 Paper: $\omega_C^n$ is very ample $n\geq 3$
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 2, 2014 at 14:25 | answer | added | Puzzled | timeline score: 1 | |
| Nov 30, 2012 at 15:04 | vote | accept | HNuer | ||
| Nov 30, 2012 at 15:04 | history | bounty ended | HNuer | ||
| Nov 30, 2012 at 14:57 | answer | added | HNuer | timeline score: 3 | |
| Nov 25, 2012 at 15:11 | comment | added | HNuer | Thank you so much Ulrich. If you turn these 3 comments into an answer I can give you the bounty points! Again thanks. | |
| Nov 25, 2012 at 13:52 | comment | added | naf | For 3), it is not claimed that the projective bundle is trivializable for all families of stable curves. | |
| Nov 25, 2012 at 10:32 | comment | added | naf | The reason for considering $n=2$ is Serre duality: to show $H^0(C,\omega^3)$ is constant one needs to show that $H^1(C,\omega^3)$ vanishes and this is equivalent to showing that $H^0(C, \omega^{-2})$ vanishes. | |
| Nov 25, 2012 at 10:15 | comment | added | naf | Let $A'$ be the ring $k[x] \times k[y]$, $A$ the subring consisting of pairs of polynomials having the same value at $0$ and $m$ the maximal ideal of $A$ consisting of pairs of polynomials vanishing at $0$. Given an element $a$ of $A'$, multiplication by $a$ gives an $A'$-linear map from $m$ to $A$. Conversely, given such a map $\phi$ , $\phi(x,0)$ is an element of $A'$ which must be of the form $(xf(x),0)$ for some element $f \in k[x]$. Similarly, $\phi(0,y) = (0,yg(y))$ for some $g \in k[y]$. So $\phi \mapsto (f,g) \in A'$ gives the desired inverse. | |
| Nov 23, 2012 at 16:51 | history | bounty started | HNuer | ||
| Nov 21, 2012 at 17:00 | comment | added | HNuer | I was pretty sure it was incorrect since it wasn't providing the correct answer :). Any thoughts on how to show it correctly? | |
| Nov 21, 2012 at 4:20 | comment | added | naf | Re 1): $Hom(\mathcal{O}_x, \mathcal{L})$ injects into $Hom(\mathfrak{m}_x, \mathcal{L})$ so it is clear that what you compute cannot be correct. | |
| Nov 20, 2012 at 19:57 | history | asked | HNuer | CC BY-SA 3.0 |