Exotic elliptic algebras of dimension 4

Abstract

Let E be an elliptic curve defined over an algebraically closed field k whose characteristic is not 2 or 3. Let τ be a translation automorphism of E that is not of order 2. In a previous paper we studied an algebra A=A(E,τ) that depends on this data: A(E,τ)=(S(E,τ)⊗M2(k))Γ where S(E,τ) is the 4-dimensional Sklyanin algebra associated to (E,τ), M2(k) is the ring of 2×2 matrices over k, and Γ is (Z/2)×(Z/2) acting in a particular way as automorphisms of S and M2(k). The action of Γ on S is compatible with the translation action of the 2-torsion subgroup E[2] on E. Following the ideas and results in papers of Artin–Tate–Van den Bergh, Smith–Stafford, and Levasseur–Smith, this paper examines the line modules, point modules, and fat point modules, over A, and their incidence relations. The right context for the results is non-commutative algebraic geometry: we view A as a homogeneous coordinate ring of a non-commutative analogue of P3 that we denote by Projnc(A). Point modules and fat point modules determine “points” in Projnc(A). Line modules determine “lines” in Projnc(A). Line modules for A are in bijection with certain lines in P(A1⁎)≅P3 and therefore correspond to the closed points of a certain subscheme L of the Grassmannian G(1,3). Shelton–Vancliff call L the line scheme for A. We show that L is the union of 7 reduced and irreducible components, 3 quartic elliptic space curves and 4 plane conics in the ambient Plücker P5, and that deg⁡(L)=20. The union of the lines corresponding to the points on each elliptic curve is an elliptic scroll in P(A1⁎). Thus, the lines on that elliptic scroll are in natural bijection with a corresponding family of line modules for A.