A Conjecture of Mukai Relating Numerical Invariants of Fano Manifolds

A complex manifold X of dimension n such that the anticanonical bundle -K (X) := det TX is ample is called a Fano manifold. Besides the dimension, other two integers play an essential role in the classification of these manifolds, namely the pseudoindex of X, i (X) , which is the minimal anticanonical degree of rational curves on X, and the Picard number rho (X) , the dimension of N (1)(X), the vector space generated by irreducible complex curves modulo numerical equivalence . A (generalization of a) conjecture of Mukai says that rho (X) (i (X) - 1) a parts per thousand currency sign n. In this paper we present some partial steps towards the conjecture, we show how one can interpretate and possibly solve it with the use of families of rational curves on a uniruled variety, and more generally with the instruments of Mori theory. We consider also other related problems: the description of some Fano manifolds which are at the border of the Mukai relations and how the pseudoindex changes via (some) birational transformation.