Abstract
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.
Citation
Kento FUJITA. "On a Generalization of the Mukai Conjecture for Fano Fourfolds." Tokyo J. Math. 37 (2) 319 - 333, December 2014. https://doi.org/10.3836/tjm/1422452796
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