ABSTRACT | A rational curve C in a nonsingular variety X is standard if under the normalization f:P1”ęC, the vector bundle f?T(X) decomposes as O(2)+O(1)p+Oq for some nonnegative integers satisfying p+q=dimX?1. For a Fano manifold X of Picard number 1, a general rational curve of minimal degree through a general point is standard. It has been asked whether all rational curves of minimal degree through a general point are standard. In a joint work with Hosung Kim, we find a negative example to this question. Let d>2 be an odd integer and let f(x1,...,xn,xn+1), n>d?1, be a weighted homogeneous polynomial of degree 2d with respect to the weights wt(x1)=...=wt(xn)=1 and wt(xn+1)=2. Let Xf be a Veronese double cone of dimension n associated to a general choice of f. This is an n-dimensional Fano manifold of Picard number 1 with index n+2?d. We study the variety of minimal rational tangents at a general point x of Xf, the projective variety defined as the union of the tangent directions to rational curves of minimal degree through x. We show that the normalization morphism of the variety of minimal rational tangents is not an immersion if 2d