Slopes of smooth curves on Fano manifolds
Jun-Muk Hwang, Hosung Kim, Yongnam Lee, and Jihun Park

TL;DR
This paper classifies when Fano manifolds are slope stable with respect to smooth curves, especially focusing on rational curves, and finds that most are stable except for projective space.
Contribution
It provides a complete classification of slope stability for Fano manifolds concerning smooth curves, highlighting the special case of projective space.
Findings
Fano manifolds of dimension ≥3 are mostly slope stable with respect to smooth curves.
Fano threefolds with Picard number 1 are slope stable unless they are projective space.
The case of rational curves is particularly significant in the stability analysis.
Abstract
Ross and Thomas introduced the concept of slope stability to study K-stability, which has conjectural relation with the existence of constant scalar curvature K\"ahler metric. This paper presents a study of slope stability of Fano manifolds of dimension with respect to smooth curves. The question turns out to be easy for curves of genus and the interest lies in the case of smooth rational curves. Our main result classifies completely the cases when a polarized Fano manifold is not slope stable with respect to a smooth curve. Our result also states that a Fano threefold with Picard number 1 is slope stable with respect to every smooth curve unless is the projective space.
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