Let R = k[x, y] be the polynomial ring over an algebraically closed field k. Let Tbe a sequence of nonnegative integers that occurs as the Hilbert function of a length-nArtinian quotient of R. The nonsingular projective variety G T parametrizes all graded ideals Iof R = k[x, y] for which the Hilbert function H(R/I) = T(see Iarrobino, A. (1977). Punctual Hilbert Schemes. Mem. Amer. Math. Soc. Vol. 10, #188, Providence: American Mathematical Society). We show that G T is birational to a certain product SGrass(T) of small Grassmann varieties (Proposition 3.15), and that over k = ℂ the birational map induces an additive ℤ-isomorphism τ : H*(G T ) → H*(SGrass(T)) of homology groups (Theorem 3.29). The map τ is not usually an isomorphism of rings. We determine the ring H*(G T ) when where G T ⊂ ℙυ×ℙ j (Theorem 4.5). In this case G T is a desingularisation of the υ-secant bundle Sec(υ, j) of the degree jrational normal curve. We use this ring H*(G T ) to determine the number of ideals satisfying an intersection of ramification conditions at different points (Example 4.6). We also determine the classes in H*(G T ) of the pullback of the singular locus of Sec(υ, j) and of the pullbacks of the higher singular loci (Theorem 4.12). Let Ebe a monomial ideal of R, satisfying H(R/E) = T, where ∣ T ∣ = n: it corresponds to a partition P(E) of nhaving diagonal lengths T. A main tool is that the family of graded ideals having initial monomials Eis a cell 𝕍(E). We connect these cells to ramification conditions, using the Wronskian determinant, and to a “hook code” for P(E). Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.