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 bundlewhere 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.

Computer Algebra and Polynomials

“…3] (see also [DAn04,Iar13]). Also, in [CKPU13], a very interesting study of how the μ-basis of a parametrization having generic μ (μ = d/2 ) and very singular points looks like has been made.…”

“…It should be also mentioned that in the recently paper [Iar13], an attempt of the stratification proposed in Problem 2 for this kind of curves is done, but only with respect to the value of μ and no further parameters.…”

Moving Curve Ideals of Rational Plane Parametrizations

“…so the role of the µ-basis is crucial to understand K. Indeed, any minimal set of generators of K contains a µ-basis, so the pairs (1, µ), (1, d − µ) are always elements of B(K). The study of the geometry of V d according to the stratification done by µ has been done in [CSC98, Section 3] (see also [DAn04,Iar13]). Also, in [CKPU13], a very interesting study of how the µ-basis of a parametrization having generic µ (µ = d/2 ) and very singular points looks like has been made.…”

“…It should be also mentioned that in the recently paper [Iar13], an attempt of the stratification proposed in Problem 2 for this kind of curves is done, but only with respect to the the value of µ and no further parameters.…”

Irreducible components of Hilbert schemes of rational curves with given normal bundle

“…It is well known that the classification of rational curves by the splitting type of f * T P s produces irreducible subvarieties of H rat d,s ; see [Ver83,Ram90]. One can also look at [AR15] for a geometric characterization of rational curves with a given splitting of f * T P s and at [Iar14] for related results in the commutative algebra language.…”