Rational curves on homogeneous cones
  Let G/Q be an homogeneous variety embedded in a projective space P thanks to an ample line bundle L. Take a projective space containing P and form the cone X over G/Q, we call this a cone over an homogeneous variety.   Let $\alpha$ a class of 1-cycle on X. In this article we describe the irreducible components of the scheme of morphisms of class $\alpha$ from a rational curve to X.   The situation depends on the line bundle L : if the projectivised tangent space to the vertex contains lines (i.e. if G/Q contains lines in P) then the irreducible components are described as in our paper math.AG/0407123 by the difference between Cartier and Weil divisors. On the contrary if there is no line in the projectivised tangent space to the vertex then there are new irreducible components corresponding to the multiplicity of the curve through the vertex.   As in math.AG/0407123 we use a resolution Y of X (the blowing-up) and study the curves on Y. 
Nicolas Perrin