Strata of vector spaces of forms in R = 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 ⊂ R j having dimension d. The strata Grass H (R j , d) ⊂ G determined by the Hilbert functions H = H(R/(V)) or, equivalently, by the Betti numbers of the algebras R/(V), are locally closed and irreducible of known dimension. They satisfy a frontier property that the closure of a stratum is its union with lower strata [I1, I2]. The strata are determined also by the decomposition of the restricted tangent bundle $$\mathcal{T}_V = \varphi _V^* (\mathcal{T})$$ where $$\mathcal{T}$$ is the tangent bundle on ℙ d −1 and ϕ V is the rational curve determined by V. 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] and studied further in [KPU], that are determined in part by singularities of ϕ 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 and another of C. D’Andrea concerning the dimension and closure of μ families of parametrized rational curves from planar [CSC, D] to higher dimensional embeddings (Theorem 1.11).