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Ross and Thomas introduced the concept of slope stability to study  K ‐stability, which has conjectural relation with the existence of constant scalar curvature Kähler metric. This paper presents a study of slope stability of Fano manifolds of dimension  n ⩾3 with respect to smooth curves. The question turns out to be easy for curves of genus at least 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 three‐fold  X  with Picard number 1 is slope stable with respect to every smooth curve unless  X  is the projective space.

Slopes of smooth curves on Fano manifolds