Bounds on the Picard rank of toric Fano varieties with minimal curve constraints

TL;DR

This paper investigates bounds on the Picard rank of smooth toric Fano varieties with minimal rational curves, proving new cases of a conjecture using toric Mori theory and combinatorics.

Contribution

It establishes bounds on Picard ranks for certain toric Fano varieties and proves new cases of a conjecture relating to minimal rational curves.

Findings

Proved a version of the Chen-Fu-Hwang conjecture in high dimensions.

Established new bounds for Picard ranks in high-degree cases.

Utilized toric Mori theory and Fano polytope combinatorics effectively.

Abstract

We study the Picard rank of smooth toric Fano varieties possessing families of minimal rational curves of given degree. We discuss variants of a conjecture of Chen-Fu-Hwang and prove a version of their statement that recovers the original conjecture in sufficiently high dimension. We also prove new cases of the original conjecture for high degrees in all dimensions. Our main tools come from toric Mori theory and the combinatorics of Fano polytopes.