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].

In this paper we classify graded reflexive ideals, up to isomorphism and shift, in certain threedimensional Artin-Schelter regular algebras. This classification is similar to the classification of right ideals in the first Weyl algebra, a problem that was completely settled recently. The situation we consider is substantially more complicated however.  2004 Elsevier Inc. All rights reserved.

We classify reflexive graded right ideals, up to isomorphism and shift, of generic cubic three-dimensional Artin-Schelter regular algebras. We also determine the possible Hilbert functions of these ideals. These results are obtained by using similar methods as for quadratic Artin-Schelter algebras [K. De Naeghel, M. Van den Bergh, Ideal classes of three-dimensional Sklyanin algebras, J. Algebra 276 (2) (2004) 515-551; K. De Naeghel, M. Van den Bergh, Ideal classes of three dimensional Artin-Schelter regular algebras, J. Algebra 283 (1) (2005) 399-429]. In particular our results apply to the enveloping algebra of the Heisenberg-Lie algebra from which we deduce a classification of right ideals of the invariant ring A ϕ 1 of the first Weyl algebra A 1 = k x, y /(xy − yx − 1) under the automorphism ϕ(x) = −x, ϕ(y) = −y.

The Hilbert scheme of n points in the projective plane has a natural stratification obtained from the associated Hilbert series. In general, the precise inclusion relation between the closures of the strata is still unknown. Guerimand, Ph.D Thesis, Universite'de Nice, 2002 studied this problem for strata whose Hilbert series are as close as possible. Preimposing a certain technical condition he obtained necessary and sufficient conditions for the incidence of such strata. In this paper we present a new approach, based on deformation theory, to Guerimand's result. This allows us to show that the technical condition is not necessary.

We characterize the Hilbert functions and minimal resolutions of (critical) Cohen-Macaulay graded right modules of Gelfand-Kirillov dimension two over generic quadratic and cubic three-dimensional ArtinSchelter regular algebras. See also [Y. Berest, G. Wilson, Ideal classes of the Weyl algebra and noncommutative projective geometry (with an appendix by Michel Van den Bergh), Int. Math. Res. Not. 2002 (26) (2002) 1347-1396; L. Le Bruyn, Moduli spaces for right ideals of the Weyl algebra, J. Algebra 172 (1995) 32-48; T.A. Nevins, J.T. Stafford, Sklyanin algebras and Hilbert schemes of points, math.AG/0310045, 2003. [8,14,15]].