Summary: FANO MANIFOLDS WITH LONG EXTREMAL RAYS
MARCO ANDREATTA, GIANLUCA OCCHETTA
Abstract. Let X be a Fano manifold of pseudoindex iX whose Picard number
is at least two and let R be an extremal ray of X with exceptional locus Exc(R).
We prove an inequality which bounds the length of R in terms of iX and of
the dimension of Exc(R) and we investigate the border cases.
In particular we classify Fano manifolds X of pseudoindex iX obtained blowing
up a smooth variety Y along a smooth subvariety T such that dim T < iX .
A smooth complex projective variety of dimension n is called Fano if its anticanon-
ical bundle -KX = n
TX is ample. The index of X, rX, is the largest natural
number m such that -KX = mH for some (ample) divisor H on X, while the
pseudoindex iX is defined as the minimum anticanonical degree of rational curves
on X and it is an integral multiple of rX.
The pseudoindex is related to the Picard number X of X by a conjecture which
claims that X(iX - 1) n, with equality if and only if X (PiX -1
; this con-
jecture appeared in  as a generalization of a similar one (with the index in place