The symmetric representation of lines in PG(F³⊗F³)

Let F be a finite field, an algebraically closed field, or the field of real numbers. Consider the vector space V=F³⊗F³ of 3 × 3 matrices over F, and let G≤PGL(V) be the setwise stabiliser of the corresponding Segre variety S_3,3(F) in the projective space PG(V). The G-orbits of lines in PG(V) were determined by the first author and Sheekey as part of their classification of tensors in F²⊗V in [15]. Here we solve the related problem of classifying those line orbits that may be represented by symmetric matrices, or equivalently, of classifying the line orbits in the F-span of the Veronese variety V₃(F)⊂S_3,3(F) under the natural action of K=PGL(3,F). Interestingly, several of the G-orbits that have symmetric representatives split under the action of K, and in many cases this splitting depends on the characteristic of F. Although our main focus is on the case where F is a finite field, our methods (which are mostly geometric) are easily adapted to include the case where F is an algebraically closed field, or the field of real numbers. The corresponding orbit sizes and stabiliser subgroups of K are also determined in the case where F is a finite field, and connections are drawn with old work of Jordan and Dickson on the classification of pencils of conics in PG(2,F), or equivalently, of pairs of ternary quadratic forms over F.