Some remarks on the study of good contractions |
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Authors: | M Andreatta |
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Institution: | (1) Dipartimento di Matematica, Universitá di Trento, 38050 Povo (TN), Italia |
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Abstract: | LetX be a complex projective variety with log terminal singularities admitting an extremal contraction in terms of Minimal Model
Theory, i.e. a projective morphism φ:X→Z onto a normal varietyZ with connected fibers which is given by a (high multiple of a) divisor of the typeK
x+rL, wherer is a positive rational number andL is an ample Cartier divisor. We first prove that the dimension of anu fiberF of φ is bigger or equal to (r-1) and, if φ is birational, thatdimF≥r, with the equalities if and only ifF is the projective space andL the hyperplane bundle (this is a sort of “relative” version of a theorem of Kobayashi-Ochiai). Then we describe the structure
of the morphism φ itself in the case in which all fibers have minimal dimension with the respect tor. If φ is a birational divisorial contraction andX has terminal singularities we prove that φ is actually a “blow-up”. |
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Keywords: | 14E30 14J40 14C20 14J45 |
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