Timeline for A question on the topological change of dualizing a SLAG fibration.
Current License: CC BY-SA 3.0
4 events
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| May 30, 2013 at 12:37 | comment | added | Tony Pantev | The fact that the minimal resolution of the compactified relative Picard is again a K3 can be checked directly from Kodaira's classification of singular fibers. The type of the singular fiber is determined by the local monodromy and the local monodromy doesn't change under dualization because it is in $SL_2$. Once you know that singular fibers are the same as the original ones, the canonical class formula for an elliptic surface tells you that you have a K3. This doesn't work in higher dimensions already topologically as Mark explained above. | |
| May 29, 2013 at 21:59 | comment | added | Jason Starr | @Tobias: The compactified relative Picard automatically has a section. For a genus 1 fibration of a K3 surface over $\mathbb{P}^1$, there is a topological obstruction to the existence of even a continuous section (e.g., if there is a multiple fiber). Thus the topology of the fibration can change. The fact that the compactified relative Picard is again a K3 surface is a special case of a general theory of Mukai about moduli spaces of (semi)stable sheaves on hyper-Kaehler manifolds (although there may be a more direct proof for K3s). | |
| May 29, 2013 at 20:11 | comment | added | Tobias | Thank you for the answer. I implicitly assume that there exists a section, but does it matter whether or not we have a section? As to the 3rd line, why is the new family again K3 surface? Why doesn't your argument work in dimension three? Thank you for your help. | |
| May 29, 2013 at 18:45 | history | answered | Tony Pantev | CC BY-SA 3.0 |