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A characterization of products of projective spaces. (English) Zbl 1115.14034
By a conjecture proposed by Mukai in 1988, for a Fano manifold $$X$$ covered by rational curves always should take place the inequality $$\rho_X(r_X-1) \leq \dim X$$, where $$\rho_X$$ is the rank of $$\operatorname{Pic}X$$ over $${\mathbb Z}$$, and $$r_X$$ is the index of $$X$$, defined as the greatest integer $$m$$ such that $$-K_X = mL$$ for some $$L \in \operatorname{Pic}X$$. A recent progress in this direction is marked in the paper [L. Bonavero, C. Casagrande, O. Debarre and S. Druel, Comment. Math. Helv. 78, No. 3, 601–626 (2003; Zbl 1044.14019)] where the conjecture of Mukai is verified for $$\dim X \leq 4$$ and for certain special Fano manifolds of greater dimensions [see also J. Wisniewski, Manuscr. Math. 68, No. 2, 135–141 (1990; Zbl 0715.14033); M. Andreatta, E. Chierici and G. Occhetta, Cent. Eur. J. Math. 2, No. 2, 272–293 (2004; Zbl 1068.14049)]. The main result of the present paper is the following:
Theorem 1.1. A smooth complex projective variety $$X$$ of dimension $$n$$ is isomorphic to a product of projective spaces $${\mathbb P}^{n(1)} \times\dots \times {\mathbb P}^{n(k)}$$ if and only if there exist $$k$$ unsplit covering families of rational curves $$V^1,\dots,V^k$$ of degrees $$n(1)+1,\dots,n(k)+1$$ with $$\Sigma n(i) = n$$ such that the numerical classes of $$V^1,\dots, V^k$$ are linearly independent in $$N_1(X)$$.

##### MSC:
 14J45 Fano varieties 14J40 $$n$$-folds ($$n>4$$)
##### Keywords:
Fano variety; conjecture of Mukai
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