Strata of vector spaces of forms in k[x,y] and of rational curves in P^k

Abstract

Consider the polynomial ring R=k[x,y] over an infinite field k and the subspace R_j of degree-j homogeneous polynomials. The Grassmanian G=Grass (R_j,d) parametrizes the vector spaces V in R_j having dimension d. The strata Grass_H(R_j,d) in G determined by the Hilbert function H=H(R/(V)) or, equivalently, by the Betti numbers of the algebra R/(V), are locally closed and irreducible of known dimension. We had previously shown also that they satisfy a frontier property that the closure of a stratum is its union with lower strata [I1,I2]. Each stratum corresponds to a partition, and the the poset of strata under closure is isomorphic to the poset of corresponding partitions in the Bruhat order. They are coarser than the strata defined by D. Cox, A. Kustin, C. Polini, and B.Ulrich [CKPU] recently that are determined in part by singularities of the rational curve determined by V. We explain these results and give examples to make them more accessible. We also generalize a result of D. Cox, T. Sederberg and F. Chen [CSC] and another of C. D'Andrea [D] concerning the dimension and closure of families of parametrized rational curves from planar to higher dimensional embeddings.