 
Summary: MINIMAL MODEL PROGRAM WITH SCALING AND
ADJUNCTION THEORY
MARCO ANDREATTA
Abstract. 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 KX +rL
for high rational number r. For this we run a Minimal Model Program with
scaling relative to the divisor KX + rL. We give some applications, namely
the classification up to birational equivalence of quasi polarized pairs with
sectional genus 0, 1 or 2 and of embedded projective varieties X PN with
degree smaller than 2codimPN (X) + 2.
1. Introduction
Let X be a complex projective normal variety of dimension n and L be a nef and
big line bundle on X. The pair (X, L) is called a quasi polarized pair. The
goal of Adjunction Theory is to classify quasi polarized pairs via the study of the
positivity of the adjunction divisors KX + rL, with r a positive rational number.
This has been done extensively in the case in which L is ample, i.e. (X, L) is a
polarized pair; [BS95] is the best account on this case .
However the set up of quasi polarized pairs is certainly more natural: in particular
when passing to a resolution of the singularities and taking the pull back of L. The
