A bound of lengths of chains of minimal rational curves on Fano manifolds of Picard number 1

TL;DR

This paper establishes a sharp bound on the minimal number of minimal rational curves needed to connect two general points on Fano manifolds with Picard number 1, providing explicit calculations for dimensions less than 8.

Contribution

It introduces a fundamental bound on chain lengths of minimal rational curves on Fano manifolds of Picard number 1, with explicit computations for low-dimensional cases.

Findings

Established a sharp bound for chain lengths on Fano manifolds

Computed chain lengths explicitly for dimensions less than 8

Provided a fundamental argument applicable to these bounds

Abstract

In this paper, we investigate the minimal length of chains of minimal rational curves needed to join two general points on a Fano manifold of Picard number 1. In particular, we give a sharp bound of the length by a fundamental argument. As an application, we compute the length for Fano manifolds of dimension < 8.