Quadratic Algebras Associated with the Union of a Quadric and a Line in [formula omitted

Abstract

We define a family of graded quadratic algebras A σ (on 4 generators) depending on a fixed nonsingular quadric Q in P 3, a fixed line L in P 3 and an automorphism σ ∈ Aut( Q ∪ L). This family contains O q ( M 2( C )), the coordinate ring of quantum 2 × 2 matrices. Many of the algebraic properties of A σ are shown to be determined by the geometric properties of { Q ∪ L, σ}. For instance, when Aσ = O q ( M 2( C )), then the quantum determinant is the unique (up to a scalar multiple) homogeneous element of degree 2 in O q ( M 2( C )) that vanishes on the graph in P 3 × P 3 of σ| Q but not on the graph of σ| L. Following results of M. Artin, J. Tate, and M. Van den Bergh ("The Grothendieck Festschrift," Birkhäuser, Basel, 1990; and Invent. Math. 106, 1991, 335-388), we study point and line modules over the algebras A σ, and find that their algebraic properties are consequences of the geometric data. In particular, the point modules are in one-to-one correspondence with the points of Q or L, and the line modules are in bijection with the lines in P 3 that either lie on Q or meet L. In the case of O q ( M 2( C )), when q is not a root of unity, the quantum determinant annihilates all the line modules M( l) corresponding to lines l ⊂ Q; the determinant generates the whole annihilator for such l ⊂ Q if and only if l and L = ∅.