Tokyo Journal of Mathematics

On a Generalization of the Mukai Conjecture for Fano Fourfolds


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Let $X$ be a complex $n$-dimensional Fano manifold. Let $s(X)$ be the sum of $l(R)-1$ for all the extremal rays $R$ of $X$, the edges of the cone $\operatorname{NE}(X)$ of curves of $X$, where $l(R)$ denotes the minimum of $(-K_X \cdot C)$ for all rational curves $C$ whose classes $[C]$ belong to $R$. We show that $s(X)\leq n$ if $n\leq 4$. And for $n\leq 4$, we completely classify the case the equality holds. This is a refinement of the Mukai conjecture on Fano fourfolds.

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Tokyo J. Math., Volume 37, Number 2 (2014), 319-333.

First available in Project Euclid: 28 January 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J45: Fano varieties
Secondary: 14E30: Minimal model program (Mori theory, extremal rays) 14J35: $4$-folds


FUJITA, Kento. On a Generalization of the Mukai Conjecture for Fano Fourfolds. Tokyo J. Math. 37 (2014), no. 2, 319--333. doi:10.3836/tjm/1422452796.

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