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.

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

Stable rank-2 vector bundles on ℙ³ with c₁=0 and c₂=3.

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).

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

On etudie les stratifications du schema de Hilbert des courbes lisses rationnelles de l'espace projectif de dimension trois par le fibre normal et par le fibre tangent restreint

Sur le schéma de Hilbert des courbes rationnelles de P3

We extend the Shatz stratification of sheaves to arbitrary families of projective schemes. This allows a stratification of Hilbert schemes. We investigate how the Harder-Narasimhan polygon of the restriction of the tangent bundle ΘIPn to space curves reflects the geometry of these curves and their embeddings.

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Georg Hein

Papers

Curves in ${\bf P}^3$ with Good Restriction of the Tangent Bundle