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I am looking for an example of computation of the isomorphism classes of $n$-point modules over a non-commutative generated graded algebra (assuming all good properties such as Noetherian property). Recall that $n$-point module $M$ over graded $k$-algebra $A$ is defined to be a graded module generated in degree $0$ with Hilbert polynomial $H_M(t)=\frac{n}{1-t}$, i.e. $\dim_{k}M_d=n$. Of course, when $A$ is commutative, the isomorphism classes of $n$-point modules form the Hilbert scheme of $n$ points over $\mathrm{Proj}(A)$.

I have seen some computation of the isomorphism classes of $1$-point modules but not 2- or higher over strictly non-commutative algebra. Could anyone kindly give me a reference for non-trivial computation? Or show me some simple computation?

Thank you very much in advance.

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1 Answer 1

You could look at the paper "Linear Modules over Sklyanin Algebras" by Joanna Staniszkis. In this paper she calculates all the linear modules over a Sklyanin algebra on any number of generators. She also constructs all the possible short exact sequences involving linear modules.

The method is very geometric. Denote the $n$-dimensional Sklyanin algebra by $A_n(E,\tau)$ where $E$ is the associated (smooth) elliptic curve and $\tau$ a point on it. The $d$-dimensional linear modules correspond either to $d$-secants (so hyperplanes in projective space that intersect the elliptic in $d+1$ points) or so-called $d$-singular planes.

To see this coincides with the computations in previous papers for the smaller linear modules, one can see from Corollary 3.5(a) loc. cit. that the point modules for $n=4$ correspond to points on the elliptic curve and 4 singular points, which are the singular loci of the singular quadrics containing $E$.

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