A point model for the free cyclic submodules over ternions (Q351106)

For a commuative field \(F\), the ring \(T\) of upper triangular \(2\times 2\) matrices over \(F\) ist called the ring of ternions. It is shown that the set of all free cyclic submodules of \(T^2\) admits a point model in \(\mathrm{PG}(7,F)\) which is a smooth algebraic variety \(\mathcal{X}\cup \mathcal{Y}\), where \(\mathcal X\) corresponds to the unimodular submodules and \(\mathcal Y\) (corresponding to the non-unimodular ones) is a line.