Let X be a Fano variety of dimension n, pseudoindex i X and Picard number ρX. A generalization of a conjecture of Mukai says that ρX(i X−1)≤n. We prove that the conjecture holds for a variety X of pseudoindex i X≥n+3/3 if X admits an unsplit covering family of rational curves; we also prove that this condition is satisfied if ρX> and either X has a fiber type extremal contraction or has not small extremal contractions. Finally we prove that the conjecture holds if X has dimension five.
@article{bwmeta1.element.doi-10_2478_BF02476544,
author = {Marco Andreatta and Elena Chierici and Gianluca Occhetta},
title = {Generalized Mukai conjecture for special Fano varieties},
journal = {Open Mathematics},
volume = {2},
year = {2004},
pages = {272-293},
zbl = {1068.14049},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_BF02476544}
}
Marco Andreatta; Elena Chierici; Gianluca Occhetta. Generalized Mukai conjecture for special Fano varieties. Open Mathematics, Tome 2 (2004) pp. 272-293. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_BF02476544/
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