Slopes of smooth curves on Fano manifolds

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 $n\geq 3$ with respect to smooth curves. The question turns out to be easy for curves of genus $\geq 1$ and the interest lies in the case of smooth rational curves. Our main result classifies completely the cases when a polarized Fano manifold $(X, -K_X)$ is not slope stable with respect to a smooth curve. Our result also states that a Fano threefold $X$ with Picard number 1 is slope stable with respect to every smooth curve unless $X$ is the projective space.