Timeline for Zero-cohomology of birational varieties
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 18, 2013 at 17:05 | vote | accept | Joaquín Moraga | ||
| Feb 19, 2013 at 0:16 | |||||
| Feb 18, 2013 at 17:05 | vote | accept | Joaquín Moraga | ||
| Feb 18, 2013 at 17:05 | |||||
| Feb 18, 2013 at 16:23 | comment | added | Sasha | @Mohan: Yes, I assumed implicitly that $Y$ is normal. On the other hand, note that the higher direct images of the structure sheaf (which may be nonzero if the singularity of $Y$ is not rational) do not change the global sections. | |
| Feb 18, 2013 at 16:10 | comment | added | Sándor Kovács | @Mohan: why is it irrelevant in the small case? If $Y$ is regular in codimension 2, but not normal, then $H^0(X,f^*\mathscr O_Y)\neq H^0(Y,\mathscr O_Y)$. | |
| Feb 18, 2013 at 15:16 | comment | added | Mohan | The projection formula does not give what you say in the first line. It only gives $h^0(X,f^*L)=h^0(Y,L\otimes f_*\mathcal{O}_X$. So, for example if $X$ is the normalizatiion of a non-normal $Y$, we do not get what you say. Of course this is irrelevant in the `small' case above. | |
| Feb 18, 2013 at 8:36 | comment | added | Sándor Kovács | This seems way too complicated. You don't need anything else but the definition of $f_*$.... | |
| Feb 18, 2013 at 5:49 | vote | accept | Joaquín Moraga | ||
| Feb 18, 2013 at 17:05 | |||||
| Feb 18, 2013 at 5:47 | history | answered | Sasha | CC BY-SA 3.0 |