Generalizing a question of Mukai, we conjecture that a Fano manifold X with Picard number and pseudo-index satisfies ( - 1) <= dim(X). We prove this inequality in several situations: X is a Fano manifold of dimension <= 4, X is a toric Fano manifold of dimension <= 7 or X is a toric Fano manifold of arbitrary dimension with >= dim(X) / 3 + 1. Finally, we offer a new approach to the general case.
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Laurent Bonavero, Stéphane Druel, Olivier Debarre, Cinzia Casagrande, Sur une conjecture de Mukai. Comment. Math. Helv. 78 (2003), no. 3, pp. 601–626DOI 10.1007/S00014-003-0765-X