Partially Symmetric Tensors: Connections and Classifications

In this talk, we present an interesting correspondence between partially symmetric 3×3×r tensors over Fq (for r≤6), linear systems of conics in PG(2,q), and subspaces of PG(5,q). We review the history of classifying these linear systems in PG(2,q) up to projective equivalence, an open problem dating back to 1908. We outline recent advances, showing how properties of the quadric Veronesean in PG(5,q) can be utilized to identify a set of complete invariants for projectively inequivalent pencils and webs of conics in PG(2,q).  These results contribute to the classification of partially symmetric 3x3xr tensors over Fq under the action of the group stabilising rank-1 tensors.