Classifying subspaces of cube-central order 3 matrices using the geometry of cubic curves (Q2891070)

Let \(U\) be the subspace of trace zero matrices of the algebra \(A\) of order 3 matrices with coefficients in a separably closed field \(F\) of characteristic different from 2 and 3. A matrix \(x\in A\) is called cube-central if \(x^3\in F\). Let \(V\) be a 3-dimensional subspace of \(U\) consisting of cube-central elements. Then it follows that \(x^3= \text{det}(x)\) for all \(x\in V\) and hence each such vector space \(V\) gives rise to a ternary cubic form \(f_V\) mapping \(V\) into \(F\) by means of \(f_V(x)= x^3\). Such a \(V\) is called a cubic subspace of \(A\). It is now natural to try to classify up to conjugacy those cubic subspaces \(V\) of \(A\) for which \(f_V\) is nonsingular, i.e., to give, for each conjugacy class, vectors which span a representative of the class.NEWLINENEWLINE The main result of this paper is an explicit answer to this problem. The proof depends essentially on particular geometric properties of nonsingular cubic curves.