An Abridged Exposition of Mori’s Bend and Break Argument
نویسنده
چکیده
Mori’s “bend and break” argument was originally introduced in 1979 in order to prove Hartshorne’s conjecture, showing that (at least in characteristic 0) one can recognize projective spaces as precisely the smooth projective varieties with ample tangent bundles (here for a vector bundle E to be ample means that if P(E) is the associated projective bundle, OP(E)(1) is ample). He then applied the same arguments shortly thereafer in his groundbreaking work on the structure of threefolds. Mori’s argument stands out as one of the most spectacular instances of proving a theorem for complex varieties using reduction to positive characteristic, and we begin by stating and outlining the proof of his result. Recall the following terminology: a rational curve is an integral curve with normalization isomorphic to P; a Fano variety is a smooth projective variety with −KX ample; and a divisor D is nef if and only if D ·C ≥ 0 for every curve C ⊆ X.
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