Minimal Model Program with scaling and adjunction theory

Let (X,L) be a quasi polarized pairs, i.e. X is a normal complex projective variety and L is a nef and big line bundle on it. We study, up to birational equivalence, the positivity (nefness) of the adjoint bundles K_X + rL for high rational number r. For this we run a Minimal Model Program with scaling relative to the divisor K_X +rL. We give some applications, namely the classification up to birational equivalence of quasi polarized pairs with sectional genus 0,1 and of embedded projective varieties X < P^N with degree smaller than 2codim(X) +2.Comment: 12 pages. Proposition 3.6 of the previous version was incomplete. Some proofs have been shortened. The paper will be published on International Journal of Mathematic

Local Fano-Mori contractions of high nef-value

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Morphisms of Projective Varieties from the viewpoint of Minimal Model Theory

In this Lecture Notes we present, in a sufficiently self contained way, our contributions and interests in the field of Minimal Model Theory. We study Fano-Mori spaces, both from the biregular and the birational point of view. For the former we recall and develop Kawamata's Base Point Free technique and some of Mori's deformation arguments. For the latter we lean on Sarkisov and #-Minimal Model Programs. In writing these notes we want to give our point of view on this area of research. We are not trying to give a treatment of the whole subject. These notes collect some topics we presented in three mini-courses which were held in Wykno (Pl) (1999), Recife (Br) (2000) and Ferrara (It) (2000), respectively.Comment: 78 pages, AMSLaTeX2e, to appear on Dissertationes Mathematicae, Polish Ac. Sc. Warsa