Fano manifolds which are not slope stable along curves

Abstract

We show that a Fano manifold $(X,-K_{X})$ is \textit{not} slope stable with respect to a smooth curve $Z$ if and only if $(X,Z)$ is isomorphic to one of (projective space, line), (product of projective line and projective space, fiber of second projection) or (blow up of projective space along linear subspace of codimension two, nontrivial fiber of blow up).