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 \(5\).


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