Line scheme

A line module over a connected graded algebra \(A\) is a cyclic graded module \(M = \bigoplus_{i \geq 0} M_i\) with Hilbert series \(\operatorname{h}_M(t) = (1-t)^{-2}\), i.e. \(\dim_k M_i = i + 1\) — the module-theoretic analogue of a line \(\mathbb{P}^1 \subset \mathbb{P}^3\), one dimension up from a point module.

The line modules of \(A\) are parametrised by a projective scheme, the line scheme of \(A\). A line of \(\mathbb{P}^3 = \mathbb{P}(A_1^*)\) is a point of the Grassmannian \(\operatorname{Gr}(2,4)\), which sits via its Plücker embedding as the Klein quadric in \(\mathbb{P}^5\); the line scheme is the closed subscheme of \(\operatorname{Gr}(2,4)\) cut out by the conditions for a line to carry a line module. Shelton–Vancliff proved this functor is representable and, for a quadratic Auslander-regular algebra of global dimension 4, gave the explicit determinantal equations used here (the construction implemented as lineSchemeFourDim in the Macaulay2 package AssociativeAlgebras).

For the commutative polynomial ring every line of \(\mathbb{P}^3\) is a line module, so the line scheme is all of \(\operatorname{Gr}(2,4)\) — the Klein quadric itself, of dimension \(4\). For a generic quadratic \(\mathbb{P}^3\) it is instead a curve of degree \(20\), and Shelton–Vancliff show that once the line scheme is one-dimensional it already determines the defining relations of the algebra. So a line scheme of higher dimension is the mark of a genuinely special algebra:

• the Sklyanin algebra (and its degenerate cousin \(\mathrm{S}_\infty\), which shares its point scheme) has a two-dimensional line scheme: the surface of secant lines to the quartic elliptic curve \(E\) that is its point scheme — an elliptic ruled surface of degree \(8\). Its line modules are exactly the lines of \(\mathbb{P}^3\) meeting \(E\) with multiplicity \(\geq 2\) (Shelton–Vancliff; see also Smith);
• Vancliff’s algebra, whose point scheme is a quadric surface, has a three-dimensional line scheme;
• many double-Ore families have a line scheme that is a plane (a pencil of lines through a point, or the lines in a plane) together with a curve.

Overview

The line scheme of every family on this site, computed at a generic point of its parameter space (over \(\mathbb{Q}(\text{params})\) where feasible, otherwise over \(\mathrm{GF}(p)\) at generic parameters for two primes \(p \equiv 1 \bmod 12\); the dimension and degree agree across both). The dimension is the key invariant: \(1\) is generic, higher is special.