QUADRATIC ALGEBRAS ASSOCIATED WITH THE UNION OF A QUADRIC AND A LINE IN P(3)

[摘要] We define a family of graded quadratic algebras A(sigma) (on 4 generators) depending on a fixed nonsingular quadric Q in P3, a fixed line L in P3 and an automorphism sigma is-an-element-of Aut(Q or L). This family contains O(q)(M2(C)), the coordinate ring of quantum 2 x 2 matrices. Many of the algebraic properties of A(sigma) are shown to be determined by the geometric properties of {Q or L, sigma}. For instance, when A(sigma) = O(q)(M2(C)), then the quantum determinant is the unique (up to a scalar multiple) homogeneous element of degree 2 in O(q)(M2(C)) that vanishes on the graph in P3 x P3 of sigma\Q but not on the graph of sigma\L. Following results of M. Artin, J. Tate, and M. Van den Bergh (The Grothendieck Festschrift,'' Birkhauser, Basel, 1990; and Invent. Math. 106, 1991, 335-388), 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 or L, and the line modules are in bijection with the lines in P3 that either lie on Q or meet L. In the case of O(q)(M2(C)), when q is not a root of unity, the quantum determinant annihilates all the line modules M(l) corresponding to lines l subset-of Q; the determinant generates the whole annihilator for such l subset-of Q if and only if l and L = empty set. (C) 1994 Academic Press, Inc.

[发布日期] 1994-04-01