Alignment Correspondences

φ : R→̃ÔX,p with φ(I1, . . . , Ir) = (a1, . . . , ar)} and its closure C(I1, . . . , Ir) in the appropriate product of Hilbert schemes. If the ideals Ij are monomial, we will say that the space C(I1, . . . , Ir) is an alignment correspondenceswith interior U(I1, . . . , Ir). The significance of the Ij’s being monomial is that in this case we can show that the space U(I1, . . . , Ir) in most cases is an affine bundle on the flag bundle on X and in the remaining cases has an étale covering by such a space and give a classification of these spaces via measuring sequences (Theorem 3.1). While many interiors of alignment correspondences are naturally isomorphic while the alignment correspondences themselves may vary quite a bit. In particular, we show that in some there is a compactification of an isomorphism class of interiors of alignment correspondences dominating all alignment correspondences with that interior and sometimes there is not. Among the reasons for studying these spaces are for their applications to enumerative geometry,