Rational curves and bounds on the Picard number of Fano manifolds. (English) Zbl 1208.14033

The Mukai conjecture, and its generalization, state a bound on the Picard number of Fano manifolds \(X\) in terms of its index (pseudoindex), that is the (numerical) divisibility of the canonical class [see L. Bonavero, C. Casagrande, O. Debarre and S. Druel, Comment. Math. Helv. 78, No. 3, 601–626 (2003; Zbl 1044.14019)]. The paper under review proves the conjecture under the assumption that the pseudoindex \(i_X\), is not too small, namely \(i_X\geq (\dim X+3)/3\). This is done via the study of families of rational curves on the variety \(X\). This approach allows to reprove the conjecture in dimension \(4\) and \(5\).


14J45 Fano varieties
14E30 Minimal model program (Mori theory, extremal rays)


Zbl 1044.14019
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