2014/05/21 by Marco Andreatta, Luca Tasin, Andreatta, Marco +1
Mathematics · #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG
paper · pdf · doi:10.48550/arxiv.1405.5353
11 pages. To appear in Math. Research Letters
arxiv created 2015/04/23 · arxiv updated 2015/04/24
Let X be a variety with at most terminal \mathbb Q-factorial singularities of dimension n. We study local contractions f:X→ Z supported by a \mathbb Q-Cartier divisor of the type KX+ τL, where L is an f-ample Cartier divisor and τ≥ 0 is a rational number. Equivalently, f is a Fano-Mori contraction associated to an extremal face in NE(X)KX+τL = 0; these maps naturally arise in the context of the minimal model program. We prove that, if τ> (n-3) >0, the general element X' ∈ |L| is a variety with at most terminal singularities. Then we apply this to characterize, via an inductive argument, some birational contractions as above with τ> (n-3)≥ 0.