Suppose we have an elliptic fibration $f:X\to \mathbb{P}^1$, with a singular fiber $F$, can we construct an elliptic fibration over $\mathbb{P}^1$ with fiber $nF$?
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3$\begingroup$ Compose with a degree $n$ covering of $\mathbb P^1$ by $\mathbb P^1$ that ramifies at the point having singular fiber? Or do you also want an elliptic fibration in the end? $\endgroup$– HongluCommented May 10, 2014 at 15:11
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$\begingroup$ As far as I remember, to be able to be multiple, a singular fiber should not be simply connected, i.e., it should be of type $I_p$. Otherwise, you can construct a fibration; do you want it to be related to the original one in any way? $\endgroup$– Alex DegtyarevCommented May 10, 2014 at 15:53
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$\begingroup$ @Honglu Yes I still want to have an elliptic fiberation in the end. Thanks for your clarification. $\endgroup$– user39380Commented May 11, 2014 at 2:01
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$\begingroup$ @AlexDegtyarev Thanks for your comment! Is there a reference for that? By $I_p$ do you mean the classification of minimal elliptic fiberation? Can we reduce the case that $f$ is not relatively minimal to it? $\endgroup$– user39380Commented May 11, 2014 at 2:14
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$\begingroup$ Refs are not my strongest point. This must all be due to Kodaira, but try [MR1288304] for further info. $\endgroup$– Alex DegtyarevCommented May 11, 2014 at 9:15
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1 Answer
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The answer is yes if $n$ is a prime power by a theorem of Liu-Lorenzini-Raynaud; see p. 497, Corollary 6.7 in http://www.math.u-bordeaux1.fr/~qliu/articles/LLR.pdf (This answered a question of Neron.) It's true in general if your fiber $F$ is multiplicative.
By the way: the answer is no if you are considering honest elliptic fibrations (with a section) when you say "fibration". In fact, assuming you are working with $X$ non-singular, the existence of a section forces every fiber to have at least one reduced irreducible component. Thus, it can't be non-reduced everywhere. (This was already mentioned in the comments.)
Also useful might be (from the above paper): "Examples of multiple fibers can be found in [18], Theorem 0.1, in [24], Sect.8,in[23],Sect.3, and in[48],9.4.1"
answered May 10, 2014 at 18:38