Compositio Mathematica

Fano threefolds in positive characteristic

a1 D.P.M.M.S., Cambridge University, 16 Mill Lane, Cambridge CB2 1SB, U.K.

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We show that the birational classification in positive characteristic of smooth Fano threefolds $X$ with Picard number 1 is the same as in characteristic zero. In particular, there are no exotic such Fanos; as a consequence of the classification, $X$ is shown to be liftable without ramification to characteristic zero and to contain a line. The main techniques employed are those of Ekedahl and of Mori and Takeuchi.

Key Words: Fano variety; projection; flop; extermal rays; monodromy.