On a generalization of the Mukai conjecture for Fano fourfolds

TL;DR

This paper refines the Mukai conjecture for Fano fourfolds by establishing an upper bound on the sum of extremal ray lengths and classifying cases of equality.

Contribution

It proves that s(X) ≤ n for Fano manifolds of dimension up to 4 and classifies all cases where equality holds, refining the Mukai conjecture.

Findings

s(X) ≤ n for n ≤ 4

Complete classification when s(X) = n for n ≤ 4

Refinement of the Mukai conjecture for Fano fourfolds

Abstract

Let X be a complex Fano manifold of dimension n. Let s(X) be the sum of l(R)-1 for all the extremal rays of X, the edges of the cone NE(X) of curves of X, where l(R) denotes the minimum of (-K_X \cdot C) for all rational curves C whose class [C] belongs 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.