Abstract
Let X be a projective variety with ℚ-factorial terminal singularities and let L be an ample Cartier divisor on X. We prove that if f is a birational contraction associated to an extremal ray R ⊂ N E ( X ) ¯ such that R·(KX+(n−2)L)<0, then f is a weighted blow-up of a smooth point. We then classify divisorial contractions associated to extremal rays R such that R·(KX+rL)<0, where r is a non-negative integer, and the fibres of f have dimension less than or equal to r+1.
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