Lengths of chains of minimal rational curves on Fano manifolds

TL;DR

This paper investigates the minimal number of minimal rational curves required to connect two general points on Fano manifolds with specific properties, providing new bounds and insights into their geometric structure.

Contribution

It introduces new bounds on the length of chains of minimal rational curves on certain Fano manifolds, especially in low dimensions and special structures.

Findings

Bounded the minimal length of rational chains for specific Fano manifolds.

Improved degree bounds for Fano 5-folds with Picard number 1.

Analyzed the structure of Fano manifolds with double cover structures.

Abstract

In this paper, we consider a natural question how many minimal rational curves are needed to join two general points on a Fano manifold X of Picard number 1. In particular, we study the minimal length of such chains in the cases where the dimension of X is at most 5, the coindex of X is at most 3 and X equips with a structure of a double cover. As an application, we give a better bound on the degree of Fano 5-folds of Picard number 1.