The homology groups of the variety of complete pairs $X_{13}$ of zero-dimensional subschemes of lengths 1 and 3 of projective space

Using the action of the one-dimensional diagonal torus group, we obtain the Betti numbers (the ranks of the homology groups as free modules) of the variety of complete pairs $X_{13}$ of zero-dimensional subschemes of lengths 1 and 3 of complex projective space.