 
Summary: GENERALIZED MUKAI CONJECTURE
FOR SPECIAL FANO VARIETIES
MARCO ANDREATTA, ELENA CHIERICI, AND GIANLUCA OCCHETTA
Abstract. Let X be a Fano variety of dimension n, pseudoindex iX and
Picard number X . A generalization of a conjecture of Mukai says that
X (iX  1) n. We prove that the conjecture holds if: a) X has pseu
doindex iX n+3
3
and either has a fiber type extremal contraction or does
not have small extremal contractions b) X has dimension five.
1. Introduction
Let X be a Fano variety, that is a smooth complex projective variety whose anti
canonical bundle KX is ample. We denote with rX the index of X and with iX
the pseudoindex of X, defined respectively as
rX = max{m N   KX = mL for some line bundle L},
iX = min{m N   KX · C = m for some rational curve C X}.
In 1988, Mukai [9] proposed the following conjecture:
Conjecture A. Let X be a Fano variety of dimension n. Then
X(rX  1) n.
A more general conjecture (since iX rX), which we will consider here, is the
