THE ONE-DIMENSIONAL LINE SCHEME OF A CERTAIN FAMILY OF QUANTUM P 3 S. and

Transcription

1 THE ONE-DIMENSIONAL LINE SCHEME OF A CERTAIN FAMILY OF QUANTUM P 3 S Richard G. Chandler richard.chandler@mavs.uta.edu students.uta.edu/rg/rgc7061 and Michaela Vancliff vancliff@uta.edu Department of Mathematics, P.O. Box University of Texas at Arlington, Arlington, TX Abstract. A quantum P 3 is a noncommutative analogue of a polynomial ring on four variables, and, herein, it is taken to be a regular algebra of global dimension four. It is well known that if a generic quadratic quantum P 3 exists, then it has a point scheme consisting of exactly twenty distinct points and a one-dimensional line scheme. In this article, we compute the line scheme of a family of algebras whose generic member is a candidate for a generic quadratic quantum P 3. We find that, as a closed subscheme of P 5, the line scheme of the generic member is the union of seven curves; namely, a nonplanar elliptic curve in a P 3, four planar elliptic curves and two nonsingular conics. Introduction A regular algebra of global dimension n is often viewed as a noncommutative analogue of a polynomial ring on n variables. Generalizing the language in [1], such an algebra is sometimes called a quantum P n 1. In [], quantum P s were classified according to their point schemes, with the point scheme of the most generic quadratic quantum P depicted by an elliptic curve in P. Consequently, a similar description is desired for quadratic quantum P 3 s using their point schemes or their line schemes, where the definition of line scheme was given in [11]. However, to date, very few line schemes of quadratic quantum P 3 s are known, especially of algebras that are candidates for generic quadratic quantum P 3 s. As explained in [14], if a generic quadratic quantum P 3 exists, then it has a point scheme consisting of exactly twenty distinct points and a one-dimensional line scheme. Hence, in this article, we compute the line scheme of a family 010 Mathematics Subject Classification:. 14A, 16S37, 16S38. Key words and phrases:. line scheme, point scheme, elliptic curve, regular algebra, Plücker coordinates. This work was supported in part by NSF grants DMS and DMS

2 LINE SCHEME OF A CERTAIN FAMILY OF QUANTUM P 3 S of algebras that appeared in [3, 5], and whose generic member is a candidate for a generic quadratic quantum P 3. The article is outlined as follows. Section 1 begins with some definitions, including the introduction of the family of algebras considered herein. The point schemes of the algebras are computed in Section in Proposition., whereas Sections 3 and 4 are devoted to the computation of the line scheme and identifying the lines in P 3 to which the points of the line scheme correspond. In particular, our main results are Theorems 3.1, 3.3 and 4.1. In the first two, we prove that the line scheme of the generic member is the union of seven curves; namely, a nonplanar elliptic curve in a P 3 (a spatial elliptic curve), four planar elliptic curves and two nonsingular conics. In Theorem 4.1, we find that if p is one of the generic points of the point scheme, then there are exactly six distinct lines of the line scheme that pass through p. An Appendix is provided in Section 5 that lists polynomials that are used throughout the article. It is hoped that data from the one-dimensional line scheme of any potentially generic quadratic quantum P 3 will motivate conjectures and future research in the subject. In fact, the results herein suggest that the line scheme of the most generic quadratic quantum P 3 is conceivably the union of two spatial elliptic curves and four planar elliptic curves (see Conjecture 4.). 1. The Algebras In this section, we introduce the algebras from [3, 5] that are considered in this article. Throughout the article, k denotes an algebraically closed field and M(n, k) denotes the vector space of n n matrices with entries in k. If V is a vector space, then V will denote the nonzero elements in V, and V * will denote the vector-space dual of V. In this section, we take char(k), but, in Sections 3 and 4, we assume char(k) = 0 owing to the computations in those sections. Definition 1.1. [3, 5] Let γ k and write A(γ) for the k-algebra on generators x 1,..., x 4 with defining relations: x 4 x 1 = ix 1 x 4, x 3 = x 1, x 3 x 1 = x 1 x 3 x, x 3 x = ix x 3, x 4 = x, x 4 x = x x 4 γx 1, where i = 1. By construction of A(γ) in [3], A(γ) is a regular noetherian domain of global dimension four with Hilbert series the same as that of the polynomial ring on four variables. As remarked in [3], the special member A(1) was studied in [10] and, if γ 4, then A(γ) has a finite point scheme consisting of twenty distinct points and a one-dimensional line scheme. Since the computation of the point scheme was omitted from [3], we will outline the computation of it in Section.

3 LINE SCHEME OF A CERTAIN FAMILY OF QUANTUM P 3 S 3 It should be noted that A(γ) = A( γ), for all γ k, under the map that sends x x and x k x k for all k. There also exist antiautomorphisms of A(γ) defined by ψ 1 : x 1 x 3 and x x 4, ψ : x λx 3 and x 4 λx 1, where λ k with λ 4 = γ. These latter maps will be useful in Sections 3 and 4. The reader should note that the point scheme given in [10] for A(1) has some sign errors in the formulae. Moreover, A(1) was studied in [7] in the context of finding the scheme of lines associated to each point of the point scheme. For background material on point modules, line modules, point schemes, line schemes, regular algebras and some of the historical development of the subject, the reader is referred to [14].. The Point Scheme of A(γ) In this section, we compute the point scheme of the algebras A(γ) given in Definition 1.1. Our method follows that of [], and we continue to assume that char(k) in this section. Let V = 4 i=1 kx i. Following [], we write the relations of A(γ) in the form Mx = 0, where M is a 6 4 matrix and x is the column vector given by x T = (x 1,..., x 4 ). Thus, we may take M to be the matrix x ix 1 0 x 3 ix 0 x M = 1 0 x 3 0, 0 x 0 x 4 x 3 x x 1 0 γx 1 x 4 0 x and, by [], the point scheme of A(γ) can be identified with the zero locus, p(γ), in P(V * ) of all the 4 4 minors of M. Fifteen polynomials given by these minors are listed in Section 5.1 in the Appendix. We will prove that, if γ 4, then p(γ) is finite with twenty distinct points. Let p = (α 1,..., α 4 ) p(γ). If α 1 = 0, then it is straightforward to prove that p is one of the points e = (0, 1, 0, 0), e 3 = (0, 0, 1, 0), e 4 = (0, 0, 0, 1). Thus, we assume α 1 = 1. If, in addition, α 4 = 0, then rank(m) = 0 if and only if α = 0 = α 3, so we obtain the point e 1 = (1, 0, 0, 0). Hence, we may assume α 1 = 1 and α 4 0. With this assumption, a computer-algebra program such as Wolfram s Mathematica yields three polynomials that determine the remaining closed points in p(γ): ρ 1 = x 8 4 4x γ, ρ = x 3 ix 3 x 4 1, ρ 3 = γx ix x 3 x 5 4. (In fact, 5.1.1, 5.1. and evaluated at x 1 = 1 generate the other polynomials in Section 5.1 evaluated at x 1 = 1, and determine ρ 1, ρ, ρ 3.) Since ρ 1 = 0 if and only if (x 4 4 ) = 4 γ, we find that ρ 1 has eight distinct zeros if and only if γ 4; if γ = 4, then ρ 1 has exactly four

4 4 LINE SCHEME OF A CERTAIN FAMILY OF QUANTUM P 3 S distinct zeros, each of multiplicity two. Given a zero x 4 to ρ 1, the equation ρ = 0 has a unique solution for x 3 if and only if x 4 4 = 4, but this implies ρ 1 0 as γ 0, which is false; hence ρ has two distinct zeros for all γ k. The following remark will be useful in the proof of Proposition.. Remark.1. (cf., [14]) If the zero locus z of the defining relations of a quadratic algebra on four generators with six defining relations is finite, then z consists of twenty points counted with multiplicity. Proposition.. Let A(γ) and p(γ) be as above and let Z γ ρ 1, ρ, ρ 3 in P(V * ). (a) For every γ k, p(γ) = {e 1,..., e 4 } Z γ. (b) If γ 4, then p(γ) has exactly twenty distinct points. denote the scheme of zeros of (c) If γ = 4, then p(γ) has exactly twelve distinct points; the eight closed points of Z γ have multiplicity two in p(γ) and the remaining four points of p(γ) each have multiplicity one. (d) For every γ k, the closed points in P(V * ) P(V * ) on which the defining relations of A(γ) vanish are given by: (e 1, e ), (e, e 1 ), (e 3, e 4 ), (e 4, e 3 ) and points of the form ( (1, α, α 3, α 4 ), (1, iα α3, α3 1, iα 4 ) ), where (1, α, α 3, α 4 ) Z γ and i = 1. Proof. The preceding discussion proves that if γ 4, then the number of distinct closed points in p(γ) is twenty, so, by Remark.1, (b) follows. On the other hand, if γ = 4, then the zeros of ρ 1 have multiplicity two, so, counting multiplicity, the eight distinct points in Z γ have multiplicity two. Thus, each e i has multiplicity one, by Remark.1. Hence, (c) and (a) follow. Part (d) is easily verified by computation with the matrix M using polynomials 5.1.1, 5.1. and in the Appendix. Corollary.3. For all γ k, there exists an automorphism σ : p(γ) p(γ) which, on closed points, is defined by: e 1 e, e 3 e 4, σ ( (1, α, α 3, α 4 ) ) = (1, iα α 3, α 1 3, iα 4 ) for all (1, α, α 3, α 4 ) Z γ. Hence, on the closed points of p(γ), σ has two orbits of length two and n orbits of length four, where n = 4 if Z γ = 16 and n = if Z γ = 8. Proof. The fact the map exists on the closed points of p(γ) is a consequence of Proposition.(d); its existence on the scheme follows from [9, Theorem 4.1.3]. The size of the orbits may be verified by computation.

5 LINE SCHEME OF A CERTAIN FAMILY OF QUANTUM P 3 S 5 3. The Line Scheme of A(γ) In this section, we compute the line scheme L(γ) of the algebras A(γ) as a closed subscheme of P 5. Our arguments follow the method given in [1], which is summarized below in Section 3.1. In Section 3., we compute the closed points of the line scheme, and, in Section 3.3, we prove that the line scheme is a reduced scheme, and so is given by its closed points. The main results of this section are Theorems 3.1 and 3.3. Henceforth, we assume that char(k) = Method. In [1], a method was given for computing the line scheme of any quadratic algebra on four generators that is a domain and has Hilbert series the same as that of the polynomial ring on four variables. In this subsection, we summarize that method while applying it to A(γ); further details may be found in [1]. The first step in the process is to compute the Koszul dual of A(γ). This produces a quadratic algebra on four generators with ten defining relations. One then rewrites those ten relations in the form of a matrix equation similar to that used in Section ; in this case, however, it yields the equation ˆMz = 0, where z T = (z 1,..., z 4 ) (where {z 1,..., z 4 } is the dual basis in V * to {x 1,..., x 4 }) and ˆM is a 10 4 matrix whose entries are linear forms in the z i. One then produces a 10 8 matrix from ˆM by concatenating two 10 4 matrices, the first of which is obtained from ˆM by replacing every z i in ˆM by u i k, and the second is obtained from ˆM by replacing every z i in ˆM by v i k, where (u 1,..., u 4 ), (v 1,..., v 4 ) P 3. For A(γ), this process yields the following 10 8 matrix: 0 u v u v u v u v 4 0 M(γ) = u 3 0 u 1 0 v 3 0 v u 4 0 u 0 v 4 0 v. u iu 1 v iv 1 0 u 3 iu 0 0 v 3 iv 0 u 1 0 u 3 γu v 1 0 v 3 γv 0 u u 1 u 4 0 v v 1 v 4 Each of the forty-five 8 8 minors of M(γ) is a bihomogeneous polynomial of bidegree (4, 4) in the u i and v i, and so each such minor is a linear combination of products of polynomials of the form N ij = u i v j u j v i, where 1 i < j 4. Hence, M(γ) yields forty-five quartic polynomials in the six variables N ij. Following [1], one then applies the map: N 1 M 34, N 13 M 4, N 14 M 3, N 3 M 14, N 4 M 13, N 34 M 1,

6 6 LINE SCHEME OF A CERTAIN FAMILY OF QUANTUM P 3 S to the polynomials, which yields forty-five quartic polynomials in the Plücker coordinates M ij on P 5. The line scheme L(γ) of A(γ) may be realised in P 5 as the scheme of zeros of these forty-five polynomials in the M ij together with the Plücker polynomial P = M 1 M 34 M 13 M 4 +M 14 M 3. For A(γ), these polynomials were found by using Wolfram s Mathematica and are listed in Section 5. of the Appendix. In the remainder of this section, we compute and describe L(γ) as a subscheme of P 5. The lines in P(V * ) that correspond to the points of L(γ) are described in Section Computing the Closed Points of the Line Scheme. Our procedure in this subsection focuses on finding the closed points of the line scheme L(γ) of A(γ); in the next subsection, we will prove that L(γ) is reduced and so is given by its closed points. We denote the variety of closed points of L(γ) by L (γ) and the zero locus of a set S of polynomials by V(S). Subtracting the polynomials and produces M 14 M 3 M4. If M 14 = M 3 = M 4 = 0, then M 1 = 0 = M 34, so there is a unique solution in this case. This leaves six cases to consider: (I) M 14 M 3 0, M 4 = 0, (IV) M 3 0, M 14 = 0 = M 4, (II) M 3 M 4 0, M 14 = 0, (V) M 14 0, M 3 = 0 = M 4, (III) M 14 M 4 0, M 3 = 0, (VI) M 4 0, M 14 = 0 = M 3. We will outline the analysis for (I), (II), (IV) and (VI); the other cases follow from these four cases by using the map ψ 1 defined in Section 1. In applying the map ψ 1, the reader should recall that M ji = M ij for all i j. Case (I): M 14 M 3 0 and M 4 = 0. With the assumption that M 4 = 0, a computation of a Gröbner basis yields several polynomials, one of which is M13M 14 M 3. Hence, M 13 = 0, and another computation of a Gröbner basis yields several polynomials, two of which are: M 14 M 3 + M 1 M 34, M34 4 M14M 34 M3M 34 + γm 14 M 3 M34 + M14M 3, so that, in particular, M 1 M Using the first polynomial to substitute for M 14 M 3, and using the assumption that M 34 0, we find that the second polynomial vanishes if and only if M1 + M34 + γm 14 M 3 M14 M3 = 0. Another computation of a Gröbner basis yields only these polynomials, so that this case provides the component L 1 = V( M 13, M 4, M 14 M 3 + M 1 M 34, M1 + M34 + γm 14 M 3 M14 M3 ).

7 LINE SCHEME OF A CERTAIN FAMILY OF QUANTUM P 3 S 7 In Theorem 3.1, we will prove that L 1 is irreducible if and only if γ 16. Here we show that if γ = 16, then L 1 is the union of two nonsingular conics. Since A(4) = A( 4), it suffices to consider γ = 4. In fact, let α k and let Q = M1 + M34 + γm 14 M 3 M14 M3 + α(m 14 M 3 + M 1 M 34 ), and associate to Q the symmetric matrix α 0 1 α + γ 0 0 α + γ 1 0, α which has rank at most two if and only if Q factors. This happens if and only if (γ, α) = (±4, 1). It follows that if γ = 4, then Q = (M 1 M 34 + M 14 M 3 )(M 1 M 34 M 14 + M 3 ), and L 1 = L 1a L 1b, where L 1a = V( M 13, M 4, M 14 M 3 + M 1 M 34, M 1 + M 14 M 3 M 34 ), L 1b = V( M 13, M 4, M 14 M 3 + M 1 M 34, M 1 M 14 + M 3 M 34 ), and each of L 1a and L 1b is a nonsingular conic, since using the last polynomial in each case to substitute for M 1 in M 14 M 3 + M 1 M 34 yields a rank-3 quadratic form in each case. Moreover, L 1b is ψ 1 applied to L 1a. Case (II): M 3 M 4 0 and M 14 = 0. With the assumption that M 14 = 0, a computation of a Gröbner basis yields several polynomials, two of which are M 13 M 3 M4 and M 3 M 4 M34. Hence, M 13 = M 34 = 0. With these additional criteria, another computation of a Gröbner basis yields exactly three polynomials: M 1 f, M 3 f, M 4 f, where f = M1 3 M 1 M3 im 3 M4. Thus, f = 0. It follows that this case yields the irreducible component L = V( M 13, M 14, M 34, M1 3 M 1 M3 im 3 M4 ) of L (γ). Case (III): M 14 M 4 0 and M 3 = 0. This case is computed by applying ψ 1 to case (II), giving L 3 = V( M 1, M 13, M 3, M34 3 M14M 34 + im 14 M4 ). Case (IV): M 3 0 and M 14 = 0 = M 4. If, additionally, M 1 0, then M 13 = 0 and M i4 = 0 for all i = 1,, 3. It follows that

8 8 LINE SCHEME OF A CERTAIN FAMILY OF QUANTUM P 3 S M1 = M3, and so these assumptions yield a subvariety of L. Hence, we may assume that M 1 = 0. It follows that this case yields the irreducible component L 4 = V( M 1, M 14, M 4, M3M 34 + iγm13m 3 M34 3 ) of L (γ), so L 4 is ψ applied to L. Case (V): M 14 0 and M 3 = 0 = M 4. This case is computed by applying ψ 1 to case (IV), giving the irreducible component L 5 = V( M 3, M 4, M 34, M 1 M14 iγm13m 14 M1 3 ) of L (γ), which is also ψ applied to L 3. Case (VI): M 4 0 and M 14 = 0 = M 3. Using M 14 = 0 = M 3, a computation of a Gröbner basis yields several polynomials, one of which is M 1 M 34 M 13 M 4 whereas the others are multiples of M1 +M34. In particular, two of those polynomials are: M 1 M 4 (M1+M 34) and M34(M 1+M 34). It follows that M1+M 34 = 0, so that this case yields the component L 6 = L 6a L 6b of L (γ), where L 6a = V( M 14, M 3, M 1 M 34 M 13 M 4, M 1 + im 34 ), L 6b = V( M 14, M 3, M 1 M 34 M 13 M 4, M 1 im 34 ), and each of L 6a and L 6b is a nonsingular conic, since using M 1 ± im 34 to substitute for M 1 in M 1 M 34 M 13 M 4 yields a rank-3 quadratic form in each case. Moreover, L 6b is ψ 1 applied to L 6a. Having completed this analysis, we can see that the point V( M 1, M 14, M 3, M 4, M 34 ), that was found earlier, is contained in L 4 L 5 L 6. We summarize the above work in the next result. Theorem 3.1. Let L (γ) denote the reduced variety of the line scheme L(γ) of A(γ). If γ 16, then L (γ) is the union, in P 5, of the following seven irreducible components: (I) L 1 = V( M 13, M 4, M 14 M 3 + M 1 M 34, M1 + M34 + γm 14 M 3 M14 M3 ), which is a nonplanar elliptic curve in a P 3. (II) L = V( M 13, M 14, M 34, M1 3 M 1 M3 im 3 M4 ), which is a planar elliptic curve. (III) L 3 = V( M 1, M 13, M 3, M34 3 M14M 34 + im 14 M4 ), which is a planar elliptic curve. (IV) L 4 = V( M 1, M 14, M 4, M3M 34 + iγm13m 3 M34 3 ), which is a planar elliptic curve. (V) L 5 = V( M 3, M 4, M 34, M 1 M14 iγm13m 14 M1 3 ), which is a planar elliptic curve. (VIa) L 6a = V( M 14, M 3, M 1 M 34 M 13 M 4, M 1 + im 34 ), which is a nonsingular conic. (VIb) L 6b = V( M 14, M 3, M 1 M 34 M 13 M 4, M 1 im 34 ), which is a nonsingular conic.

9 LINE SCHEME OF A CERTAIN FAMILY OF QUANTUM P 3 S 9 If γ = 4, then L (γ) is the union, in P 5, of eight irreducible components, six of which are L, L 3, L 4, L 5, L 6a, L 6b (as above) and two of which are L 1a = V( M 13, M 4, M 14 M 3 + M 1 M 34, M 1 + M 14 M 3 M 34 ), L 1b = V( M 13, M 4, M 14 M 3 + M 1 M 34, M 1 M 14 + M 3 M 34 ), which are nonsingular conics. Proof. The polynomials were found in the preceding work, as was the geometric description for L 1a, L 1b, L 6a and L 6b, so here we discuss only the geometric description of the other components. (I) Write q 1 = M 14 M 3 + M 1 M 34 and q = M 1 + M 34 + γm 14 M 3 M 14 M 3 viewed in k[m 1, M 14, M 3, M 34 ]. Since q = M 1 (γ/)m 1 M 34 + M 34 ( M 14 (γ/)m 14 M 3 + M 3 modulo q 1, and since char(k), we may take the Jacobian matrix of this system of two polynomials to be the 4 matrix [ M 34 M 3 M 14 M 1 M 1 (γ/)m 34 (M 14 (γ/)m 3 ) (M 3 (γ/)m 14 ) M 34 (γ/)m 1 Assuming that all the minors are zero, we find that M 34 = M 1 (from columns one and four) and M 3 = M 14 (from columns two and three). Substituting these relations into the minor obtained from the last two columns yields either (γ ± 4)M 1 M 14 = 0 or γm 1 M 14 = 0, so M 1 M 14 = 0 (since γ(γ 16) 0). Substitution into q 1 implies that there is no solution, and so the Jacobian matrix has rank two at all points of V(q 1, q ). It follows that V(q 1, q ), viewed as a subvariety of P 3 = V(M 13, M 4 ), is reduced, and so L 1 is reduced. Following the method of the proof of [13, Proposition.5], if V(q 1, q ) is not irreducible, then there exists a point in the intersection of two of its irreducible components, and so the Jacobian matrix has rank at most one at that point, which is a contradiction. Hence, V(q 1, q ) is irreducible, and thus nonsingular since it is reduced. Moreover, its genus is = 1. It follows that V(q 1, q ) is an elliptic curve, and the same is true of L 1. (II) Viewing h = M 3 1 M 1 M 3 im 3 M 4 as a polynomial in k[m 1, M 3, M 4 ], the Jacobian matrix of h is a 1 3 matrix that has rank one at all points of V(h) (since char(k) ), so V(h) is nonsingular in P = V(M 13, M 14, M 34 ). (III), (IV), (V) These cases follow from (II) by applying ψ 1 or ψ as appropriate. ) ] Description of the Line Scheme. In this subsection, we prove that the line scheme L(γ) of A(γ) is reduced and so is given by L (γ) described in Theorem 3.1.

10 10 LINE SCHEME OF A CERTAIN FAMILY OF QUANTUM P 3 S Lemma 3.. For all γ k, the irreducible components of L(γ) have dimension one; in particular, L(γ) has no embedded points. Proof. By [3], A(γ) is a regular noetherian domain that is Auslander-regular and satisfies the Cohen-Macaulay property and has Hilbert series the same as that of the polymomial ring on four variables. Hence, by [11, Remark.10], we may apply [11, Corollary.6] to A(γ), which gives us that the irreducible components of L(γ) have dimension at least one. However, by Theorem 3.1, they have dimension at most one, so equality follows. Let X 1 denote the 11-dimensional subscheme of P(V V ) consisting of the elements of rank at most two, and, for all γ k, let X denote the 5-dimensional linear subscheme of P(V V ) given by the span of the defining relations of A(γ). By [11, Lemma.5], L(γ) = X 1 X for all γ k. Since X i is a Cohen-Macaulay scheme for i = 1,, and since dim(x 1 X ) = 1, the proof of [11, Theorem 4.3] (together with Macaulay s Unmixedness Theorem) rules out the possibility of embedded components. Theorem 3.3. For all γ k, the line scheme L(γ) is a reduced scheme of degree twenty. Proof. Let X 1 and X be as in the proof of Lemma 3., and let X = X 1 X. Since deg(x 1 ) = 0 by [8, Example 19.10], Bézout s Theorem for Cohen-Macaulay schemes ([6, Theorem III-78]) implies that deg(x) = 0. However, since L(γ) = X by [11, Lemma.5], the reduced scheme X of X is isomorphic to L (γ). Since the degrees of the irreducible components of L (γ) in Theorem 3.1 are as small as possible, deg(x ) = 0; that is, 0 = deg(x) deg(x ) 0, giving deg(x) = deg(x ). As X has no embedded points by Lemma 3., it follows that X = X, so X is a reduced scheme. Thus, L(γ) is reduced and has degree twenty since deg(l (γ)) = 0. The intersection points of the irreducible components of L(γ) are straightforward to compute and are listed in [4]. 4. The Lines in P 3 Parametrized by the Line Scheme In this section, we describe the lines in P(V * ) that are parametrized by the line scheme L(γ) of A(γ). We also describe, in Theorem 4.1, the lines that pass through any given point of the point scheme; in particular, if p is one of the generic points of the point scheme (that is, p Z γ ), then there are exactly six distinct lines of the line scheme that pass through p. Since we will use results from Section 3, we continue to assume that char(k) = 0.

11 LINE SCHEME OF A CERTAIN FAMILY OF QUANTUM P 3 S The Lines in P 3. In this subsection, we find the lines in P(V * ) that are parametrized by the line scheme. We first recall how the Plücker coordinates M 1,..., M 34 relate to lines in P 3 ; details may be found in [5, 8.6]. Any line l in P 3 is uniquely determined by any two distinct points a = (a 1,..., a 4 ) l and b = (b 1,..., b 4 ) l, and may be represented by a 4 matrix [ ] a1 a a 3 a 4 b 1 b b 3 b 4 that has rank two; in particular, the points on l are represented in homogeneous coordinates by linear combinations of the rows of this matrix. In general, there are infinitely many such matrices that may be associated to any line l in P 3, and they are all related to each other by applying row operations. The Plücker coordinate M ij is evaluated on this matrix as the minor a i b j a j b i for all i j, and the Plücker polynomial P = M 1 M 34 M 13 M 4 + M 14 M 3, given in Section 3.1, vanishes on this matrix. Moreover, V(P ) is the subscheme of P 5 that parametrizes all lines in P 3. Since dim(v ) = 4, we identify P(V * ) with P 3. By Theorem 3.3, L(γ) is given by Theorem 3.1. We continue to use the notation e j introduced in Section. (I) In this case, γ 16 and the component is L 1, which is a nonplanar elliptic curve in a P 3 (contained in P 5 ), where L 1 = V( M 13, M 4, M 14 M 3 + M 1 M 34, M1 + M34 + γm 14 M 3 M14 M3 ). It follows that any line l in P(V * ) given by L 1 is represented by a 4 matrix of the form: [ ] a1 0 a 3 0, ( ) 0 b 0 b 4 where a j, b j k for all j and a 1b + a 3b 4 γa 1 b a 3 b 4 a 1b 4 b a 3 = 0. In particular, if p l, then p = (λ 1 a 1, λ b, λ 1 a 3, λ b 4 ), for some (λ 1, λ ) P 1, such that a 1b + a 3b 4 γa 1 b a 3 b 4 a 1b 4 b a 3 = 0. It is easily verified that p lies on the quartic surface V( x 1x + x 3x 4 γx 1 x x 3 x 4 x 1x 4 x x 3 ) in P(V * ) for all (λ 1, λ ) P 1. Hence, the lines parametrized by L 1 all lie on this quartic surface in P(V * ) and are given by: V(x 3, x ± x 4 ), V(x 4, x 1 ± x 3 ), and V(x 1 αx 3, x βx 4 ) for all α, β k such that (α 1)(β 1) = γαβ. The case γ = 4 is discussed below.

12 1 LINE SCHEME OF A CERTAIN FAMILY OF QUANTUM P 3 S (II) In this case, the component is L, which is a planar elliptic curve, where L = V( M 13, M 14, M 34, M1 3 M 1 M3 im 3 M4 ), so any line in P(V * ) given by L is represented by a 4 matrix of the form: [ ] a1 0 a 3 a , such that a 3 1 a 1 a 3 + ia 3 a 4 = 0. It follows that L parametrizes those lines in P(V * ) that pass through e and meet the planar curve V(x, x 3 1 x 1 x 3 + ix 3 x 4); this planar curve is a (nonsingular) elliptic curve since char(k) = 0. (III) In this case, the component is L 3, which may be obtained as ψ 1 applied to L. Hence, L 3 parametrizes those lines in P(V * ) that pass through e 4 and meet the planar elliptic curve V(x 4, x 3 3 x 1x 3 + ix 1 x ). (IV) In this case, the component is L 4, which may be obtained as ψ applied to L. Hence, L 4 parametrizes those lines in P(V * ) that pass through e 3 and meet the planar elliptic curve V(x 3, x 3 4 x x 4 + iγx 1x ). (V) In this case, the component is L 5, which may be obtained as ψ 1 applied to L 4. Hence, L 5 parametrizes those lines in P(V * ) that pass through e 1 and meet the planar elliptic curve V(x 1, x 3 x x 4 + iγx 3x 4 ). (VI) In this case, the component is L 6 = L 6a L 6b, where L 6a = V( M 14, M 3, M 1 M 34 M 13 M 4, M 1 + im 34 ), L 6b = V( M 14, M 3, M 1 M 34 M 13 M 4, M 1 im 34 ), which are nonsingular conics. Following the argument from case (I), any line in P(V * ) given by L 6a is represented by a 4 matrix of the form: [ a1 a a 3 a 4 ] αa 1 βa βa 3 αa 4, such that α, β, a j k for all j, a 1 a = ia 3 a 4 and α β. A calculation similar to that used in (I) verifies that every point of the line lies on the quadric V(x 1 x ix 3 x 4 ). It follows that L 6a parametrizes one of the rulings of the nonsingular quadric V(x 1 x ix 3 x 4 ); namely, the ruling that consists of the lines V(δx 1 ɛx 4, δx 3 + iɛx ) for all (δ, ɛ) P 1. Since L 6b may be obtained by applying ψ 1 to L 6a, we find L 6b parametrizes one of the rulings of the nonsingular quadric V(x 3 x 4 ix 1 x ); namely, the ruling that consists of the lines V(δx 3 ɛx, δx 1 + iɛx 4 ) for all (δ, ɛ) P 1.

13 LINE SCHEME OF A CERTAIN FAMILY OF QUANTUM P 3 S 13 (Ia) and (Ib) In this case, γ = 4 and the component is L 1 = L 1a L 1b, where L 1a = V( M 13, M 4, M 14 M 3 + M 1 M 34, M 1 + M 14 M 3 M 34 ), L 1b = V( M 13, M 4, M 14 M 3 + M 1 M 34, M 1 M 14 + M 3 M 34 ), which are nonsingular conics. Following the argument from case (I), any line in P(V * ) given by L 1a is represented by a 4 matrix of the form ( ) such that a 1 b + a 1 b 4 + b a 3 = a 3 b 4. A calculation similar to that used in (I) verifies that every point of the line lies on the nonsingular quadric Q a = V( x 1 x + x 1 x 4 + x x 3 x 3 x 4 ) in P(V * ). Hence, the lines parametrized by L 1a all lie on Q a and are: V(x 3, x + x 4 ) and V(x 1 αx 3, (α + 1)x + (α 1)x 4 ) for all α k, which yields one of the rulings on the quadric Q a. Applying ψ 1 to these lines, it follows that the lines parametrized by L 1b are: V(x 1, x + x 4 ) and V(x 3 αx 1, (α 1)x + (α + 1)x 4 ) for all α k, which yields one of the rulings on the nonsingular quadric Q b = V( x 3 x 4 + x x 3 + x 1 x 4 x 1 x ). 4.. The Lines of the Line Scheme That Contain Points of the Point Scheme. In this subsection, we compute how many lines in P(V * ) that are parametrized by L(γ) contain a given point of p(γ). By [11, Remark 3.], if the number of lines is finite, then it is six, counting multiplicity; hence, the generic case is considered to be six distinct lines. The reader should note that a result similar to Theorem 4.1 is given in [7, Theorem IV..5] for the algebra A(1), but that result is false as stated (perhaps as a consequence of the sign errors in the third relation of (3) on Page 797 of [10]). Theorem 4.1. Suppose γ k, and let Z γ be as in Proposition.. (a) For any j {1,..., 4}, e j lies on infinitely many lines that are parametrized by L(γ). (b) Each point of Z γ lies on exactly six distinct lines of those parametrized by L(γ). Proof. Since (a) follows from (II)-(V) in Section 4.1, we focus on (b). Let p = (1, α, α 3, α 4 ) Z γ. It follows that α j 0 for all j. Suppose that γ 16. Let α = 1/α 3 and β = α /α 4, so (α 1)(β 1) = γαβ, by in Section 5.1. Hence, p V(x 1 αx 3, x βx 4 ), which is a line that corresponds to an element of L 1. Clearly, no other line given by L 1 contains p. Let r = (1, 0, α 3, α 4 ) and let l denote the line through e and r. By 5.1.9, we have 1 α3 + iα 3 α4 = 0, so r V(x, x 3 1 x 1 x 3 + ix 3 x 4). Thus, l corresponds to an element of L,

14 14 LINE SCHEME OF A CERTAIN FAMILY OF QUANTUM P 3 S and p l. Conversely, let r = (b 1, 0, b 3, b 4 ) V(x, x 3 1 x 1 x 3 + ix 3 x 4). If p lies on the line through r and e, then there exist (λ 1, λ ) P 1 such that p = (λ 1 b 1, λ, λ 1 b 3, λ 1 b 4 ). Thus, λ 1 b 1 0 and α i = b i /b 1 for i = 3, 4. Hence, r = (b 1, 0, b 1 α 3, b 1 α 4 ) = (1, 0, α 3, α 4 ) = r. It follows that no other line given by L contains p. Let r 4 = (1, α, α 3, 0) and let l 4 denote the line through e 4 and r 4. By 5.1., we have α 3 3 α 3 + iα = 0, so r 4 V(x 4, x 3 3 x 1x 3 + ix 1 x ). Thus, l 4 corresponds to an element of L 3, and p l 4. An argument similar to that of L proves that no other line given by L 3 contains p. Let r 3 = (1, α, 0, α 4 ) and let l 3 denote the line through e 3 and r 3. By 5.1.5, we have α 3 4 α α 4 + iγα = 0, so r 3 V(x 3, x 3 4 x x 4 + iγx 1x ). Thus, l 3 corresponds to an element of L 4, and p l 3. An argument similar to that of L proves that no other line given by L 4 contains p. Let r 1 = (0, α, α 3, α 4 ) and let l 4 denote the line through e 1 and r 1. By 5.1.8, we have α 3 α α 4 + iγα 3α 4 = 0, so r 1 V(x 1, x 3 x x 4 + iγx 3x 4 ). Thus, l 4 corresponds to an element of L 5, and p l 4. An argument similar to that of L proves that no other line given by L 5 contains p. By 5.1.1, we have α = ±iα 3 α 4, so either p V(x 1 x ix 3 x 4 ) or p V(ix 1 x x 3 x 4 ) (but not both, since α 3 α 4 0). In the first case, p V(α 4 x 1 x 4, α 4 x 3 + ix ) and, in the second, p V(α 4 x 1 x 4, iα 4 x 3 + x ). These lines correspond to elements of L 6a and L 6b respectively. Since each quadric has only two rulings, and since each irreducible component of L 6 parametrizes only one of the rulings in each case, no other line given by L 6 contains p. If, instead, γ = 4, the only adjustment to the above reasoning is in the case of the lines parametrized by L 1. Since γ = 4, the polynomial factors, so that is, (α + α 4 + α α 3 α 3 α 4 )(α α 4 α α 3 α 3 α 4 ) = 0, ( ) ( (1 + α3 )α + (1 α 3 )α 4 )( (1 α3 )α (1 + α 3 )α 4 ) = 0, which provides exactly two lines (of those parametrized by L 1 ) that could contain p. These lines are V(x 1 (1/α 3 )x 3, ((1/α 3 ) + 1)x + (1/α 3 ) 1)x 4 ) and V(x 3 α 3 x 1, (α 3 1)x + (α 3 + 1)x 4 ), which correspond to elements of L 1a and L 1b respectively. If the first factor of ( ) is zero, then p belongs to the first line, whereas if the second factor of ( ) is zero, then p belongs to the second line. If both factors of ( ) are zero, then α = α 3 α 4, which forces α 3 α 4 = 0, by 5.1.1, and this contradicts p Z γ. It follows that p belongs to exactly one line of those parametrized by L 1.

15 LINE SCHEME OF A CERTAIN FAMILY OF QUANTUM P 3 S 15 For all γ k, it is a straightforward calculation to show that the six lines found above are distinct. Considering Theorems 3.1, 3.3 and 4.1 in the case where γ 16, we arrive at the following conjecture. Conjecture 4.. The line scheme of the most generic quadratic quantum P 3 is isomorphic to the union of two spatial (irreducible and nonsingular) elliptic curves and four planar (irreducible and nonsingular) elliptic curves. (Here, spatial elliptic curve means a nonplanar elliptic curve that is contained in a subscheme of P 5 that is isomorphic to P 3.) This conjecture is motivated by the idea that the generic points of the point scheme should have exactly six distinct lines of the line scheme passing through each of them, with each line coming from exactly one component of the line scheme. Moreover, if the component L 6 of the line scheme L(γ) of A(γ) had not split into two smaller components, then it would likely have been a spatial elliptic curve. 5. Appendix In this section, we list the polynomials that define p(γ) and L(γ) Polynomials Defining the Point Scheme. The following are the polynomials that define the point scheme viewed as p(γ) P(V * ) of A(γ) that are given by the fifteen 4 4 minors of the matrix M in Section ; they are used in Section and in the proof of Theorem 4.1: x 1x + x 3x 4, x 1 (x 3 3 x 1x 3 + ix 1 x ), x (x 3 3 x 1x 3 + ix 1 x ), x 4 (x 3 3 x 1x 3 + ix 1 x ), x 1 (x 3 4 x x 4 + iγx 1x ), x (x 3 4 x x 4 + iγx 1x ), x 3 (x 3 4 x x 4 + iγx 1x ), x 1 (x 3 x x 4 + iγx 3x 4 ), x (x 3 1 x 1 x 3 + ix 3 x 4), iγx 1x 3 x 1x x 4 x x 3x 4, ix x 4 x 1 x x 3 x 1 x 3 x 4,

16 16 LINE SCHEME OF A CERTAIN FAMILY OF QUANTUM P 3 S x 3 1x 4 + γx 1x x 3 x 1 x 3x 4 + ix x 3 x 4, x 3 x 3 + γx 1 x x 4 x x 3 x 4 + iγx 1x 3 x 4, iγx 3 1x 3 + γx 1x x 1 x x 3 x 4 + ix 3 x 4, x 1x x x 3 γx 1 x x 3 x 4 x 1x 4 + x 3x 4, where i = 1 and γ k. 5.. Polynomials Defining the Line Scheme. The following are the forty-six polynomials in the M ij coordinates from Section 3 that define the line scheme L(γ) of A(γ): P = M 1 M 34 M 13 M 4 + M 14 M 3, M 13 M 14 M 3 M 4, 5... M 1 (γm 13 M 14 M 3 + im 1 M 14 M 4 + im 3 M 4 M 34 ), M 1 (γm 13 M 14 M 3 im 1 M 14 M 4 im 3 M 4 M 34 ), M 13 (γm 13 M 14 M 3 + im 1 M 14 M 4 + im 3 M 4 M 34 ), M 13 (γm 13 M 14 M 3 im 1 M 14 M 4 im 3 M 4 M 34 ), M 13 (γm 13 M 14 M 3 + im 1 M 14 M 4 im 3 M 4 M 34 ), M 14 (γm 13 M 14 M 3 + im 1 M 14 M 4 + im 3 M 4 M 34 ), M 3 (γm 13 M 14 M 3 + im 1 M 14 M 4 + im 3 M 4 M 34 ), M 3 (γm 13 M 14 M 3 im 1 M 14 M 4 im 3 M 4 M 34 ), M 4 (γm 13 M 14 M 3 + im 1 M 14 M 4 + im 3 M 4 M 34 ), M 34 (γm 13 M 14 M 3 + im 1 M 14 M 4 + im 3 M 4 M 34 ), M 1 (M 1 M 13 M 3 + M 13 M 14 M 34 + im 14 M 3 M 4 ), M 1 (M 1 M 13 M 3 + M 13 M 14 M 34 im 14 M 3 M 4 ), M 13 (M 1 M 13 M 3 + M 13 M 14 M 34 + im 14 M 3 M 4 ), M 14 (M 1 M 13 M 3 + M 13 M 14 M 34 + im 14 M 3 M 4 ), M 14 (M 1 M 13 M 3 + M 13 M 14 M 34 im 14 M 3 M 4 ), M 3 (M 1 M 13 M 3 + M 13 M 14 M 34 + im 14 M 3 M 4 ), M 4 (M 1 M 13 M 3 + M 13 M 14 M 34 + im 14 M 3 M 4 ), M 4 (M 1 M 13 M 3 + M 13 M 14 M 34 im 14 M 3 M 4 ), M 4 (M 1 M 13 M 3 M 13 M 14 M 34 + im 14 M 3 M 4 ), M 34 (M 1 M 13 M 3 + M 13 M 14 M 34 + im 14 M 3 M 4 ),

17 LINE SCHEME OF A CERTAIN FAMILY OF QUANTUM P 3 S M13M 3 M 4 + M 13 M 14 M3 M 13 M 14 M34 + im 14 M 3 M 4 M 34, M1M 13 M 3 + im 1 M 14 M 3 M 4 M13M 14 M 4 M 13 M14M 3, iγm 1 M 13 M3 γm 14 M3M 4 M 1 M 14 M 4 M 34 M 3 M 4 M34, iγm 13 M 14 M 3 M 34 M 13 M 14 M4 M14M 3 M 4 + M 3 M 4 M34, iγm 1 M 13 M 14 M 3 M1M 14 M 4 + M 13 M 3 M4 + M 14 M3M 4, γm 13 M14M 3 + M 1 M 13 M 3 M 34 + im 1 M14M 4 + M 13 M 14 M34, γm14m 3 + M1M 14 M 3 + M 1 M14M 34 + M 1 M3M 34 + M 14 M 3 M34, iγm 1 M13M 3 + γm 13 M 14 M 3 M 4 + M1M 13 M 4 + im 1 M 14 M4 + M 13 M 4 M34, iγm13m M1M im1m 14 M 4 M 1 M 13 M14 + M13M 4 M 34, γm 1 M 13 M 14 M 3 + M1M 14 M 4 M 1 M 13 M3 M 13 M 14 M 3 M 34 im 13 M 3 M4, iγm13m 14 M 34 + M1M 13 M 4 + im 1 M 14 M4 M 13 M14M 4 + M 13 M 4 M34, iγm1m 13 M 3 γm 1 M 14 M 3 M 4 iγm13m 14 M 4 +M 1 M14M 4 +M 14 M 3 M 4 M 34, iγm 1 M13M 3 M1M 13 M 4 + M 13 M3M 4 M 13 M 4 M34 + im 3 M4M 34, iγm1m 13 M 3 M1M M 1 M3M 4 M 13 M4M 34 + im 3 M4, γm14m 3 M 34 M 1 M14M 3 + M 1 M 3 M34 M14M im14m 4 + M 14 M34, iγm13m 3 3 γm 13 M 14 M 3 M 34 M 1 M13M 4 im 1 M 14 M 4 M 34 + M 13 M3M 34 M 13 M34, γm 1 M14M 3 + iγm13m 14 + M1M M1M 3 M 34 M 1 M14 3 M14M 3 M 34, iγm13m 3 γm 14 M3M 34 + M 1 M 14 M3 M 1 M 14 M34 + M3M 3 34 M 3 M34, iγm 1 M 14 M3 + im1m im1m 14 M 34 im 1 M3 3 im 14 M3M 34 + M3M 4, iγm 1 M 13 M 3 M 34 γm 14 M 3 M 4 M 34 M 1 M 13 M4 + M14M 4 M 34 im 14 M4 3 M 4 M34, iγm 1 M 14 M 3 M 34 im1m 14 M 3 im 1 M14M 34 M 1 M 14 M4 im 1 M3M 34 im 14 M 3 M34 + M 3 M4M 34, iγm 1 M13M 3 γm 1 M 14 M 3 M 34 iγm13m 14 M 34 + M1M 14 M 3 + M 1 M14M 34 + M 1 M3M 34 + M 14 M 3 M34, γm1m 14 M 3 +iγm 1 M13M 14 +M1 M 4 1M 14 M 1M 3 im 1 M 3 M4+M 13M 4+ M 14M 3, iγm13m 3 M 34 + γm 14 M 3 M34 + M13M 4 + M14M 3 M14M 34 + im 14 M4M 34 M3M 34 + M34, 4 where i = 1 and γ k.

18 18 LINE SCHEME OF A CERTAIN FAMILY OF QUANTUM P 3 S Acknowledgments. The authors gratefully acknowledge support from the NSF under grants DMS and DMS Moreover, the authors are grateful to B. Shelton for discussions about a potential approach towards computing the line scheme of the algebra defined in [10]; that algebra is a member of the family of algebras investigated herein. References [1] M. Artin, Geometry of Quantum Planes, in Azumaya Algebras, Actions and Modules, Eds. D. Haile and J. Osterburg, Contemporary Math. 14 (199), [] M. Artin, J. Tate and M. Van den Bergh, Some Algebras Associated to Automorphisms of Elliptic Curves, in The Grothendieck Festschrift 1, pp 33-85, Eds. P. Cartier et al, Birkhäuser Boston (1990). [3] T. Cassidy and M. Vancliff, Generalizations of Graded Clifford Algebras and of Complete Intersections, J. Lond. Math. Soc. 81 (010), (Corrigendum: 90 No. (014), ) [4] R. G. Chandler, On the Quantum Spaces of Some Quadratic Regular Algebras of Global Dimension Four, Ph.D. Thesis, University of Texas at Arlington, 016. [5] D. A. Cox, J. Little and D. O Shea, Ideals, Varieties, and Algorithms, Third Ed, Undergraduate Texts in Mathematics, Springer New York (007). [6] D. Eisenbud and J. Harris, The Geometry of Schemes, Graduate Texts in Mathematics 197, Springer-Verlag New York (000). [7] P. D. Goetz, The Noncommutative Algebraic Geometry of Quantum Projective Spaces, Ph.D. Thesis, University of Oregon, 003. [8] J. Harris, Algebraic Geometry: a First Course, Graduate Texts in Mathematics 133, Springer- Verlag New York (199). [9] L. Le Bruyn, S. P. Smith and M. Van den Bergh, Central Extensions of Three Dimensional Artin-Schelter Regular Algebras, Math. Zeitschrift (1996), [10] B. Shelton and C. Tingey, On Koszul Algebras and a New Construction of Artin-Schelter Regular Algebras, J. Algebra 41 No. (001), [11] B. Shelton and M. Vancliff, Schemes of Line Modules I, J. Lond. Math. Soc. 65 No. 3 (00), [1] B. Shelton and M. Vancliff, Schemes of Line Modules II, Comm. Alg. 30 No. 5 (00), [13] S. P. Smith and J. T. Stafford, Regularity of the Four Dimensional Sklyanin Algebra, Compositio Math. 83 No. 3 (199), [14] M. Vancliff, The Interplay of Algebra and Geometry in the Setting of Regular Algebras, in Commutative Algebra and Noncommutative Algebraic Geometry, MSRI Publications 67 (015), in press.