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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 \(5\).

MSC:

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

Citations:

Zbl 1044.14019
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References:

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