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Deformations of elliptic fiber bundles in positive characteristic. (English) Zbl 1274.14044

The paper under review concerns a question raised by Katsura and Ueno: can every elliptic surface over a field of positive characteristic be lifted to characteristic zero? Previously, non-liftable surfaces were only known to occur among quasi-elliptic surfaces and surfaces of general type. The author proves a negative answer for characteristics 2 and 3. More precisely, he proves that there are non-liftable elliptic fibre bundles, i.e., elliptic fibrations which are locally trivial in the étale topology.
The main ingredients of the construction include group actions on elliptic curves which are impossible in characteristic zero, and in particular, deformation theory. To this end, elliptic fiber bundles are considered as torsors over the relative Picard schemes. This allows the author to reduce to the Jacobian case and apply results from moduli theory for elliptic curves.
Applications include quotients of product surfaces \(E\times C\) and especially bielliptic surfaces which show several pathological phenomena such as obstructed deformations and deforming Jacobian fibrations into non-Jacobian fibrations.

MSC:

14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations
14D15 Formal methods and deformations in algebraic geometry
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References:

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