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Consider the polynomial ring R=k[x,y] over an infinite field k and thesubspace 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 strataGrass_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 andirreducible of known dimension. We had previously shown also that they satisfya frontier property that the closure of a stratum is its union with lowerstrata [I1,I2]. Each stratum corresponds to a partition, and the the poset ofstrata under closure is isomorphic to the poset of corresponding partitions inthe 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 bysingularities of the rational curve determined by V. We explain these resultsand give examples to make them more accessible. We also generalize a result ofD. Cox, T. Sederberg and F. Chen [CSC] and another of C. D'Andrea [D]concerning the dimension and closure of families of parametrized rationalcurves from planar to higher dimensional embeddings.

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