an:00589927
Zbl 0837.16023
Vancliff, M.
Quadratic algebras associated with the union of a quadric and a line in \(\mathbb{P}^ 3\)
EN
J. Algebra 165, No. 1, 63-90 (1994).
0021-8693
1994
j
16S35 14A22 17B37 14H37 16W50 14M07
graded quadratic algebras; nonsingular quadrics; coordinate rings of quantum \(2\times 2\) matrices; quantum determinants; point modules; line modules
Summary: The author defines a family of graded quadratic algebras \(A_\sigma\) (on 4 generators) depending on a fixed nonsingular quadric \(Q\) in \(\mathbb{P}^3\), a fixed line \(L\) in \(\mathbb{P}^3\) and an automorphism \(\sigma \in \text{Aut} (Q \cup L)\). This family contains \({\mathcal O}_q (M_2 (\mathbb{C}))\), the coordinate ring of quantum \(2 \times 2\) matrices. Many of the algebraic properties of \(A_\sigma\) are shown to be determined by the geometric properties of \(\{Q \cup L, \sigma\}\). For instance, when \(A_\sigma={\mathcal O}_q(M_2 (\mathbb{C}))\), then the quantum determinant is the unique (up to a scalar multiple) homogeneous element of degree 2 in \({\mathcal O}_q (M_2 (\mathbb{C}))\) that vanishes on the graph in \(\mathbb{P}^3 \times \mathbb{P}^3\) of \(\sigma|_Q\) but not on the graph of \(\sigma|_L\). Following results of \textit{M. Artin, J. Tate}, and \textit{M. Van den Bergh} [``The Grothendieck Festschrift'', Prog. Math. 86, 33-85 (1990; Zbl 0744.14024); and Invent. Math. 106, 335- 388 (1991; Zbl 0763.14001)], we study point and line modules over the algebras \(A_{\sigma}\), 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 \(Q \cup L\), and the line modules are in bijection with the lines in \(\mathbb{P}^3\) that either lie on \(Q\) or meet \(L\). In the case of \({\mathcal O}_q (M_2 (\mathbb{C}))\), when \(q\) is not a root of unity, the quantum determinant annihilates all the line modules \(M(l)\) corresponding to lines \(l \subset Q\); the determinant generates the whole annihilator for such \(l \subset Q\) if and only if \(l \cap L=\emptyset\).
0744.14024; 0763.14001