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Summary: A Conjecture of Mukai Relating
Numerical Invariants of Fano Manifolds
Marco Andreatta
Abstract. A complex manifold X of dimension n such that the anti-
canonical bundle -KX := det TX is ample is called a Fano manifold. Be-
sides the dimension, other two integers play an essential role in the clas-
sification of these manifolds, namely the pseudoindex of X, iX, which
is the minimal anticanonical degree of rational curves on X, and the
Picard number X, the dimension of N1(X), the vector space generated
by irreducible complex curves modulo numerical equivalence . A (gen-
eralization of a) conjecture of Mukai says that X(iX - 1) 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.
Mathematics Subject Classification (2000). Primary 14J45; Secondary
14E30.
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