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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 $$\text{ Pic}(X)$$.

##### MSC:
 14J45 Fano varieties 14E30 Minimal model program (Mori theory, extremal rays) 14M25 Toric varieties, Newton polyhedra, Okounkov bodies
##### Keywords:
Fano varieties; Mori theory; toric varieties
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