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 $K\le \mathrm{P}\mathrm{G}\mathrm{L}(6,q)$, $K\cong \mathrm{P}\mathrm{G}\mathrm{L}(3,q)$, stabilising the Veronese surface. We also determine a complete set of geometric and combinatorial invariants for each of the orbits.