Timeline for Examples for curve not 1-connected but $h^0(C, O_C)=1$
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 8, 2014 at 14:35 | comment | added | user39380 | Ah..you are right, I am sorry..Thanks for your patience! | |
| Apr 8, 2014 at 14:33 | comment | added | abx | I don't understand the notation $O_{-F}$. Aren't you confusing $\mathcal{O}_X(F)$, which is an invertible line bundle on $X$, with $\mathcal{O}_F$, the structure sheaf of $F$, supported on $F\subset X$? | |
| Apr 8, 2014 at 14:24 | comment | added | user39380 | Then $O_{-F}$ and $O_{-2E}$ are not isomorphic? Since they are the dual line bundle of $O_{F}$ and $O_{2E}$..$O_F\otimes_{O_X} O_{-F}=O_X$...? | |
| Apr 8, 2014 at 14:10 | comment | added | abx | $\mathcal{O}_F$ and $\mathcal{O}_{2E}$ are certainly not isomorphic, they have different support (namely $F$ and $E$). | |
| Apr 8, 2014 at 14:06 | comment | added | user39380 | Since we used $H^1(O_{-F})=H^1(O_{-2E})$ in the prove above, I was thinking there is no difference if we apply it to the 0th cohomology group. Maybe I had faute with understanding something? | |
| Apr 8, 2014 at 14:02 | comment | added | user39380 | Since we have $F$ is linear equivalent to $2E$, $O_F$ and $O_{2E}$ define the same invertible sheaf, can we argue $H^0(O_{2E})=k$ directly by this? | |
| Apr 8, 2014 at 12:53 | comment | added | abx | Yes and yes ... | |
| Apr 8, 2014 at 12:31 | comment | added | user39380 | Is that because $H^0(O_X)=H^0(O_F)=k$? Also is pencil here just mean the existence of an elliptic fibration? | |
| Apr 8, 2014 at 12:18 | comment | added | abx | 1) Every Enriques surface has such a pencil, look at any book on surfaces (e.g. Barth etc.). 2) The linear system $|2E|$ is a pencil of elliptic curves; take a smooth $F\in |2E|$. Since $H^1(X,\mathcal{O}_X)=0$, the exact sequence $0\rightarrow \mathcal{O}_X(-F)\rightarrow \mathcal{O}_X\rightarrow \mathcal{O}_F\rightarrow 0$ gives $H^1(X,\mathcal{O}_X(-F))=0$. | |
| Apr 8, 2014 at 12:13 | comment | added | user39380 | Thanks for your answer! But how do you get the example? Also I find it hard to check the $H^1(X,O_X(-2E))=0$, is there an easy way to show that? | |
| Apr 2, 2014 at 10:04 | vote | accept | CommunityBot | ||
| Apr 2, 2014 at 6:17 | history | answered | abx | CC BY-SA 3.0 |