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Fano threefolds in positive characteristic. (English) Zbl 0889.14021

The author proves that the birational classification in positive characteristic of smooth Fano three-folds X with Picard number 1 is the same as in the case of characteristic zero; in particular, there will be no any exotic Fano three-folds (that only appear in positive characteristic). The author also proves that such \(X\) is liftable without any ramification to characteristic zero and contains a line (in characteristic zero, this second result was first assumed when people started the classifying job and later was shown to be true in general by V. V. Shokurov). The difficulty appears in positive characteristic because the conic bundles and pencils of del Pezzo surfaces may have wild behaviour. Techniques of Ekedahl, Mori and Takeuchi are also used in the paper.

MSC:

14J45 Fano varieties
14J30 \(3\)-folds
14E05 Rational and birational maps
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