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Generalized Mukai conjecture for special Fano varieties. (English) Zbl 1068.14049
A Fano variety $$X$$ is a projective variety with ample anti-canonical bundle $$-K_X$$. For such varieties, the Picard number $$\rho$$ is strictly related to the geometry of rational curves on $$X$$. Namely, defining the pseudo-index as $$i:=\min\{m: -K_X\cdot C=m$$ for some rational curve $$C\subset X\}$$, it has been conjectured that $$\rho\leq {\dim X}/(i-1)$$ and equality implies that $$X$$ is a product of projective spaces. The conjecture has been proved by J. A. Wisniewski [Manuscr. Math. 68, 135-141 (1990; Zbl 0715.14033)], when $$i>(\dim X+2)/2$$, and by L. Bonavero, C. Casagrande, O. Debarre and S. Druel [Comment. Math. Helv. 78, No.3, 601–626 (2003; Zbl 1044.14019)], for varieties of dimension $$4$$.
In the paper under review, the authors prove that the conjecture holds for $$i\geq (\dim X+3)/3$$, provided that $$X$$ admits an unsplit covering family of rational curves, i.e. a proper component of the space of rational curves Hom$$^n_{\text{bir}}(\mathbb P^1,X)$$ mod Aut($$\mathbb P^1$$), which dominates $$X$$.
Using a careful analysis of split and unsplit families of rational curves on $$X$$, the authors are then able to prove the conjecture for Fano varieties of dimension $$5$$.

##### MSC:
 14J45 Fano varieties 14E30 Minimal model program (Mori theory, extremal rays)
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##### References:
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