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On a conjecture of Mukai. (Sur une conjecture de Mukai.) (French) Zbl 1044.14019
The authors use the following notations: \(X\) – a Fano variety, \(\rho_X\) – its Picard number, \(r_X\) – the index of \(X\), the largest integer \(m\) such that \(K_X = m \cdot L\) holds in the Picard group of \(X\), and \(\iota_X\) – the pseudo index, the smallest intersection number of the form \((-K_X)\cdot C \) with \(C\) a rational curve in \(X\).
Mukai conjectured that the inequality \(\rho_X(r_X-1) \leq \text{ dim}(X)\) holds. The authors generalize this conjecture to \[ \rho_X(\iota_X-1) \leq \text{ dim}(X) \] and prove it for the following cases:
\(X\) is a Fano variety of dimension \(\leq 4\), \(X\) is a toric variety of dimension \(\leq 7\), and \(X\) is toric and satisfies \(\iota_X \geq \frac{\text{ dim}(X)+3}{3}\).
The proof uses techniques of Mori and from the theory of toric varieties. It uses extremal contractions, chains of rational curves, families of rational curves and the pairing between \(N_{1(X)}\) and \(\text{ Pic}(X)\).

14J45 Fano varieties
14E30 Minimal model program (Mori theory, extremal rays)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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