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Special rays in the Mori cone of a projective variety
 

Summary: Special rays in the Mori cone
of a projective variety
Marco Andreatta and Gianluca Occhetta
Abstract
Let X be a smooth n­dimensional projective variety over an alge­
braically closed field k such that KX is not nef. We give a characterization
of non nef extremal rays of X of maximal length (i.e of length n - 1); in
the case of Char(k) = 0 we also characterize non nef rays of length n - 2.
1 Introduction
Let X be a smooth n­dimensional projective variety over an algebraically closed
field k of arbitrary characteristic.
We assume that KX is not nef, in particular that there exists an extremal ray
R in the cone NE(X) KX<0 .
The length of the ray R is the integer defined as l(R) = min {-KX .C : [C] # R};
the set Locus(R) is the set of closed point x # X such that there is a curve
x # C # # X with [C # ] # R.
A ray R is said to be nef if D.R # 0 for all e#ective divisors D # X, or
equivalently if Locus(R) = X; if a nef ray exists the variety X is therefore
uniruled.
The main result of the paper is the following characterization of the blow­up of

  

Source: Andreatta, Marco - Dipartimento di Matematica, Universitą di Trento

 

Collections: Mathematics