A Point Model for the Free Cyclic Submodules over Ternions

We show that the set of all (unimodular and non-unimodular) free cyclic submodules of T2, where T is the ring of ternions over a commutative field, admits a point model in terms of a smooth algebraic variety.


Introduction
The present paper is devoted to the study of the geometry of free cyclic submodules of T 2 , where T denotes the ring of ternions (upper triangular 2 × 2 matrices) over a commutative field F. In a more geometric language we refer to these submodules as ternionic points.We show that the set of all such points admits a model in terms of a smooth algebraic variety in a projective space on an 8-dimensional vector space over F. Our exposition is based on the Grassmann variety representing the planes (3-dimensional subspaces) of F 6 and the recent paper [8], where a model for the set of ternionic points in terms of planes of F 6 was given and exhibited.
From [9], the set of ternionic points splits into two orbits under the natural action of GL 2 (T ).One orbit comprises the set of unimodular points, the elements of the other orbit are called non-unimodular points.The variety representing the entire set of ternionic points accordingly splits into two parts.
The first arises by removing a single line from the variety, i.e., we obtain a quasiprojective variety in the terminology of [15].The second part is just that distinguished line.So, as regards the unimodular points, our results parallel those of A. Herzer, who developed a very general representation theory (for unimodular points only).See [5,Chapt. 11 and 12] and [10] for further details.It seems worth pointing out that all points of the distinguished line are smooth, thus giving a negative answer to the question whether nonunimodularity of a ternionic point would imply its image point being singular.
Results and notions which are used without further reference can be found, for example, in [5], [10], and [12].

Main results
Let F be a commutative field and T be the ring of ternions, i.e., the upper triangular matrices with entries a ij ∈ F. Sometimes we identify x ∈ F with the ternion xI, where I is the 2×2 identity matrix.F is the center of T and T is a three-dimensional algebra over F.
According to [9] the non-zero cyclic submodules of the free T -left module T 2 fall into five orbits under the natural action of the group GL 2 (T ).In the following we focus on two types of submodules given by a representative and An arbitrary submodule X from the orbit of X 0 given in (2) is obtained by applying S ∈ GL 2 (T ) to X 0 , i.e., X = X 0 S and will be called X-submodule in the following.Submodules of this type are free and arise from unimodular pairs of T 2 .They are the points of the projective line over T as considered in [5] and [10].This yields the general form of an X-submodule as with the constraint In the same way we obtain the general form of the Y -submodules using ( 3) with the additional condition The Y -submodules are precisely those free cyclic submodules which cannot be generated by a unimodular pair.We call them free non-unimodular points.
We identify T In the following we use the Grassmann variety G ⊂ F 20 = F 6 ∧ F 6 ∧ F 6 representing the three-dimensional subspaces of F 6 .We denote the vectors of the standard basis of F 20 by e ijk , 1 ≤ i < j < k ≤ 6, and write E ijk = Fe ijk for the respective base point of the projective coordinate system.
Every X-submodule is now represented by a point in the Grassmann variety G.We compute the Grassmann coordinates of an X-submodule with base points (8) and the non-vanishing coordinates read where again only the non-vanishing coordinate functions are given.
We want to understand the set X ∪ Y.For that end we derive the equations of X ∪ Y and show the geometric meaning of both, its parametrization and the equations as well.
At first we observe that X ∪ Y is contained in a subspace of dimension 8 which is given by the linear equations These are precisely the vanishing coordinate functions not mentioned in (9) together with two obvious identities.
Proof.First we note that for i ∈ {1, 2} and variable (u, v) the points Fq i (u, v) comprise a conic section c i in the plane spanned by E i35 , F(e i36 + e i45 ), and E i46 , whereas the points Fr(u, v) form the line spanned by E 356 and E 456 .Hence any γ(u, v) is a plane.
Given any point of X with parameters as in ( 9) and ( 5), we let u = a 22 and v = b 22 .Then one immediately reads off from (9) that this point belongs to γ(u, v).
Conversely, let G be a point of γ(u, v) which lies off the line Y.So we may assume G = Fg, where with (g 1 , g 2 , g 3 ) ∈ F 3 and (g 1 , g 2 ) = (0, 0).In order to show that the coordinates of G can be expressed as in (9)  We already know from (10) that Y is the Grassmann image of the set of all Y -modules.
The parametric representation of X and Y from ( 9) and ( 10) allows to derive equations of X ∪ Y by eliminating parameters.From now on we restrict ourselves to the 8-dimensional subspace given by Eqs.(11) and disregard the ambient 20-dimensional space.In other words, equations ( 11) are always assumed to be satisfied without further notice.We find nine quadratic equations which will be arranged in three groups.The first two equations describe quadratic cones erected on the conic sections c i mentioned in the proof of Lemma 1.These quadratic cones have five-dimensional subspaces for their vertices and six-dimensional generators.
Further we consider the quadratic equations They determine three quadratic cones on ruled quadrics.Together with the linear equations p 356 = p 456 = 0 the equations (15) describe a Segre variety S which is the product of a line and a plane, cf.[11, p. 189].In parametric form this Segre variety can be written as the set of all points with coordinates where all other coordinates are understood to be zero.
Finally the third set of quadratic equations reads Now we can prove the following result: Theorem 1.The set X ∪ Y is a smooth algebraic variety given by the equations (11), ( 14), (15), and (17).
Proof.It is easily verified that the coordinate functions of both parametric representations, namely that of X given in (9) and that of Y given in (10) annihilate Eqs. ( 11), ( 14), (15), and (17).
Hence (p i35 , p i45 , p i46 ) = (0, 0, 0) for at least one value of i.We may assume w.l.o.g. that this is the case for i = 1, whence v 1 = 0. We substitute (18) in the first equation of ( 14), divide by v 2 1 , and get the constraint u 1 u 3 − u 2 2 = 0 which reminds us of the equation of a conic section.The well-known Veronese parametrization of a conic section shows that there exists a pair (a 22 , b 22 ) ∈ F 2 \ {(0, 0)} and a constant k ∈ F \ {0} such that In order to show that X ∪ Y is smooth we compute the partial derivatives of the parametrization given in (9).Then it is a simple and straightforward calculation that the subspace spanned by the derivatives is of dimension 4 at any point of X ∪ Y, regardless of the characteristic of F.

( 9 )
This gives a parametric representation of the Grassmann image X , say, of the set of X-submodules.The parameters a 11 , a 22 , b 11 , b 22 ∈ F in (9) are subject to the restrictions given in (5), but there is no restriction on a 12 and b 12 .An easy calculation shows that the Grassmann image Y of the set comprising all Y -submodules is the line spanned by E 356 and E 456 .A parametric representation of Y is given by p 356 = a 22 (a 22 d 22 − b 22 c 22 ) and p 456 = b 22 (a 22 d 22 − b 22 c 22 ), 123 = p 124 = p 125 = p 126 = p 134 = p 156 = p 234 = p 256 = p 345 = p 346 = 0, p 236 − p 245 = 0, p 136 − p 145 = 0.