  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.
