On a Generalization of the Mukai Conjecture for Fano Fourfolds

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.