## 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)

### Keywords:

Mukai conjecture; Fano manifolds; rational curves; Picard number

Zbl 1044.14019
Full Text:

### References:

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