Ideal classes of three dimensional Artin–Schelter regular algebras

Abstract

We determine the possible Hilbert functions of graded rank one torsion free modules over three dimensional Artin-Schelter regular algebras. It turns out that, as in the commutative case, they are related to Castelnuovo functions. From this we obtain an intrinsic proof that the space of torsion free rank one modules on a non-commutative P 2 is connected. A different proof of this fact, based on deformation theoretic methods and the known commutative case has recently been given by Nevins and Stafford [Sklyanin algebras and Hilbert schemes of points, math.AG/0310045]. For the Weyl algebra it was proved by Wilson [Invent. Math. 133 (1) (1998) 1-41].

On incidence between strata of the Hilbert scheme of points on $$\mathbb{P}^{2}$$

“…Proof It has been shown in [7] that H ϕ is the moduli-space of ideals in A of projective dimension one which have Hilbert series ϕ. …”

“…. .. Remark 1.3 The number of Castelnuovo diagrams with weight n is equal to the number of partitions of n with distinct parts (or equivalently the number of partitions of n with odd parts) [7]. In loc.…”

“…For any Hilbert function ψ of degree n one defines a smooth connected subscheme [7,10] H ψ of Hilb n (P 2 ) by…”

“…This criterion can be used effectively since there are formulas for dim H ψ [7,10]. The tangent function t ϕ of a Hilbert function ϕ ∈ n is defined as the Hilbert function of I X ⊗ P 2 T P 2 , where X ∈ H ϕ is generic.…”

“…The authors became interested in the incidence problem while they were studying the deformations of the Hilbert schemes of P 2 which come from non-commutative geometry, see [16,6,7].…”