Timeline for Hilbert polynomial of structure sheaf of hypersurfaces
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Aug 22, 2018 at 1:20 | comment | added | Sándor Kovács | If $\mathscr L$ has a non-trivial global section, then there is a non-trivial embedding $\mathscr O_X\to \mathscr L$ and if $\mathscr F=\mathscr L/\mathscr O_X$, then $h^0(\mathscr F(n))\neq 0$ while $h^i(\mathscr F(n))=0$ for $n\gg 0$ and $i>0$. So, $\chi(\mathscr O_X(n))\neq \chi(\mathscr L(n))$. | |
| Aug 22, 2018 at 0:35 | history | edited | Jana | CC BY-SA 4.0 |
deleted 11 characters in body
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| Aug 22, 2018 at 0:35 | comment | added | Jana | @Mohan Sorry, I should have been a bit more specific. I will edit the question to replace Hilbert polynomial by Hilbert function. Then your example does not hold (only degree zero line bundle with non-trivial global section on a curve is the structure sheaf). | |
| Aug 22, 2018 at 0:13 | comment | added | Mohan | On a plane curve, any degree zero line bundle (and they exist if degree is at least three) has the same Hilbert polynomial as the structure sheaf, by Riemann-Roch. | |
| Aug 21, 2018 at 22:19 | history | asked | Jana | CC BY-SA 4.0 |