Elliptic three-folds II: Multiple fibres

Gross, Mark

doi:10.1090/S0002-9947-97-01845-X

Elliptic three-folds II: Multiple fibres
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by Mark Gross

Trans. Amer. Math. Soc. 349 (1997), 3409-3468

DOI: https://doi.org/10.1090/S0002-9947-97-01845-X

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Abstract:

Let $f:X\rightarrow S$ be an elliptic fibration with a section, where $S$ is a projective surface and $X$ is a projective threefold. We determine when it is possible to perform a logarithmic transformation along a closed subset $Z\subseteq S$ to obtain a new elliptic fibration $f’:X’\rightarrow S$ which now has multiple fibres along $Z$. This is done in the setting of Ogg-Shafarevich theory. We find a number of obstructions to performing such a logarithmic transformation, the very last of which takes values in the torsion part of the codimension 2 Chow group of $X$.

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