A classification of planes intersecting the Veronese surface over finite fields of even order

In this paper we contribute towards the classification of partially symmetric tensors in $\mathbb{F}_q^3\otimes S^2\mathbb{F}_q^3$, $q$ even, by classifying planes which intersect the Veronese surface $\mathcal{V}(\mathbb{F}_q)$ in at least one point, under the action of $K\leq \rm{PGL}(6,q)$, $K\cong \rm{PGL}(3,q)$, stabilising the Veronese surface. We also determine a complete set of geometric and combinatorial invariants for each of the orbits

On finite Steiner surfaces

Unlike the real case, for each q power of a prime it is possible to injectively project the quadric Veronesean of PG(5, q) into a solid or even a plane. Here a finite analogue of the Roman surface of J. Steiner is described. Such analogue arises from an embedding \sigma of PG(2, q) into PG(3, q) mapping any line onto a non-singular conic. Its image PG(2, q)\sigma has a nucleus, say T\sigma, arising from three points of PG(2, q^3 ) forming an orbit of the Frobenius collineation