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We show that a Fano manifold (X,-K_X) is not slope stable with respect to asmooth 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 secondprojection) or (blow up of projective space along linear subspace ofcodimension two, nontrivial fiber of blow up).

Fano manifolds which are not slope stable along curves