Timeline for No fixed components in the linear system of the line bundle generating $Pic(X)$
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
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| Feb 28, 2013 at 4:03 | comment | added | Sándor Kovács | @rick: If both $L$ and $L^∗$ have non-zero global sections, then they are both trivial. Or in other words a global section of $L^*$ would correspond to an effective divisor $F$, but then $F\sim −E$ and hence the effective divisor $E+F\sim 0$, but that is impossible on a projective variety. | |
| Feb 27, 2013 at 21:38 | comment | added | rick | I'm in the case $L.L=2$. From what you say if $kL\sim E$ then $h^0(X,kL)=h^0(X,E)=1$. From Riemann Roch and Serre duality i get $h^0(X,L)+h^0(X,L^*)\ge 2+\frac{kL.kL}{2}$. You say i have a contradiction because $h^0(X,L)\ge 2+k^2$. But why is $h^0(X,L^*)=0$? | |
| Feb 27, 2013 at 19:54 | comment | added | Sándor Kovács | Exactly. So you use Riemann-Roch and perhaps Kodaira vanishing for the higher cohomology and of course the fact that the Picard number is $1$. :) – | |
| Feb 27, 2013 at 13:27 | comment | added | J.C. Ottem | Yes, let's see. If $k=1$, then $H^0(X,L)=k$, but for an ample $L$, one always has $H^0(X,L)=L^2/2+2\ge 3$ since the higher cohomology vanishes. | |
| Feb 27, 2013 at 6:57 | comment | added | Sándor Kovács | I meant "@JC", not "@LC"... sorry. | |
| Feb 27, 2013 at 1:54 | comment | added | Sándor Kovács | @L.C.: That was my first thought, too, but why is it obvious that that cannot happen with, say, $k=1$? There exist ample divisors that are fixed, for instance a theta divisor on an abelian variety. You are right that this is impossible on a $K3$, but I think you need to prove that. Essentially this is what I did in my answer below.... Cheers! | |
| Feb 27, 2013 at 1:07 | comment | added | J.C. Ottem | If $|L|$ had a fixed component $E$, then in $Pic(X)$ we must have $E\sim kL$ for some $k>0$, which is clearly impossible.. | |
| Feb 27, 2013 at 0:58 | answer | added | Sándor Kovács | timeline score: 4 | |
| Feb 27, 2013 at 0:47 | history | asked | rick | CC BY-SA 3.0 |