Point scheme
A point module over a connected graded algebra \(A\) is a cyclic graded module \(M = \bigoplus_{i \geq 0} M_i\) with \(\dim_k M_i = 1\) for all \(i \geq 0\) — the module-theoretic analogue of a point of projective space.
The point modules of \(A\) are parametrised by a projective scheme, the point scheme of \(A\). Concretely it is cut out in \(\mathbb{P}(A_1^*)\) by the multilinearisations of the defining relations: a relation \(\sum c_{ij}\, x_i x_j = 0\) imposes a bilinear condition, and the point scheme is the locus where the resulting incidence has a nontrivial solution.
For a commutative polynomial ring the point scheme is all of \(\mathbb{P}^{d-1}\). In the noncommutative world it is typically much smaller, and it is the main geometric invariant distinguishing the families:
- the four-dimensional Sklyanin algebra has a quartic elliptic curve (plus four points);
- the skew polynomial ring has the six edges of the coordinate tetrahedron;
- the cocycle twists have a finite scheme of 20 points;
- Vancliff’s algebra has a quadric surface (dimension two).
Overview
The point scheme of every family on this site:
| family | dimension | point scheme |
|---|---|---|
| commutative | 3 | all of \(\mathbb{P}^3\) |
| Sklyanin | 1 | a quartic elliptic curve \(E\) and four points |
| skew | 1 | six lines, the tetrahedron skeleton \(K_4\) |
| Vancliff | 2 | a line and a quadric surface |
| Vancliff twist | 1 | five lines (\(K_4\) minus one edge) |
| Clifford | 0 | 20 points |
| central extension of Sklyanin | 1 | a twisted cubic and 8 points |
| Shelton–Tingey | 0 | 20 points |
| Cassidy–Goetz–Shelton | 1 | a line, two conics and two points |
| double Ore A | 1 | a union of 3 lines |
| double Ore B | 1 | a union of 2 lines |
| double Ore C | 1 | a regulus of 5 lines |
| double Ore D | 1 | a union of 4 lines |
| double Ore E | 1 | a union of 6 lines |
| double Ore F | 1 | a union of 2 lines |
| double Ore G | 1 | a union of 4 lines |
| double Ore H | 1 | a union of 4 lines |
| double Ore I | 1 | a union of 2 lines |
| double Ore J | 1 | a union of 6 lines |
| double Ore K | 1 | a union of 6 lines |
| double Ore L | 1 | a union of 6 lines |
| double Ore M | 1 | a union of 6 lines |
| double Ore N | 1 | a union of 2 lines |
| double Ore O | 1 | a union of 6 lines |
| double Ore P | 1 | a union of 2 lines |
| double Ore Q | 1 | a union of 6 lines |
| double Ore R | 2 | a quadric surface |
| double Ore S | 1 | a union of 2 lines |
| double Ore T | 1 | two lines and two conics |
| double Ore U | 1 | two lines and two conics |
| double Ore V | 1 | a union of 6 lines |
| double Ore W | 1 | a union of 2 lines |
| double Ore X | 1 | a union of 4 lines |
| double Ore Y | 1 | a union of 4 lines |
| double Ore Z | 1 | a union of 2 lines |
| Generalized Clifford 1 | 0 | 20 points |
| Generalized Clifford 2 | 0 | 20 points |
| Generalized Clifford 3 | 0 | 20 points |
| Ore extension of commutative | 1 | three lines and a rational normal quartic |
| Jordan | 1 | a line, a conic and a twisted cubic (a bouquet of rational normal curves) |
| \(\mathrm{S}_{d,i}\) | ? | — |
| \(\mathrm{S}_{d,i}\) twist | ? | — |
| \(\mathrm{S}_\infty\) | 1 | a quartic elliptic curve \(E\) and four points (the same curve as Sklyanin) |
| \(\mathrm{S}_\infty\) twist | 0 | 20 points |
| Sklyanin twist | 0 | 20 points |
| Kirkman R | ? | — |
| Kirkman S | ? | — |
| Kirkman T | ? | — |
| \(\mathrm{A}_5\) | ? | — |
| central extension of Sklyanin twist | ? | — |
| Caines | 0 | 13 points |
| deformed skew \((x_3x_2;\ x_4^2)\) | 1 | three lines and a conic |
| deformed skew \((x_2x_1;\ x_3x_4)\) | 1 | a union of 5 lines |
| deformed skew \((x_3x_2;\ x_4^2, x_1^2)\) | 1 | a line, two conics and two points |
| deformed skew \((x_3x_1, x_3x_2;\ x_4^2)\) | 1 | a union of 3 lines |
| deformed skew \((x_2x_1, x_3x_2;\ x_3^2, x_4^2)\) | 1 | a line and a conic |