The variety of complete pairs of zero-dimensional subschemes of an algebraic surface

We investigate the variety of zero-dimensional subschemes (that is, systems of points) and of given lengths and on a smooth projective algebraic surface . The variety is realized as the blowing-up of the direct product of Hilbert schemes of points along the incidence graph. It is proved that is naturally isomorphic to the variety of biflags , where . We also study the problem of the smoothness of . It is proved that is smooth for and an arbitrary using the Kodaira-Spenser rank map in the theory of determinantal varieties and also in the case when by means of a direct geometric consideration.