Timeline for Singular fibers of an elliptic fibered K3 surface.
Current License: CC BY-SA 3.0
3 events
| when toggle format | what | by | license | comment | |
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| Sep 3, 2012 at 1:54 | comment | added | Noam D. Elkies | Yes: a genus-1 fibration gives a rank-2 sugroup of Pic, and if there were any reducible fiber the rank would exceed 2. (I guess you're using the definition of an elliptic fibration that does not require a section, else the section together with the fiber already generate a copy of $U$ in Pic.) Likewise for K3 surfaces of higher degree: if Pic has rank 2 then there are no reducible fibers (and $m+2n=12\chi$). | |
| Sep 3, 2012 at 1:50 | comment | added | Tony Pantev | If $S$ is elliptic, then the fiber and the section generate a sublattice in $Pic(S)$ which is isomorphic to $U$. So if $S$ is elliptic and $Pic(S) \cong U(k)$ you will have $k = 1$. Did you mean to ask for $S$ to be only genus one fibered and $Pic(S) \cong U(k)$ ? | |
| Sep 3, 2012 at 0:23 | history | asked | Charls | CC BY-SA 3.0 |