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

TL;DR

This paper classifies planes intersecting the Veronese surface over finite fields of even order, under group actions, and identifies invariants for each orbit, advancing understanding of tensor classifications in this setting.

Contribution

It provides a detailed classification of planes intersecting the Veronese surface over finite fields of even order, including orbit invariants under group actions.

Findings

Classified planes intersecting the Veronese surface over finite fields of even order.

Determined geometric and combinatorial invariants for each orbit.

Enhanced understanding of tensor symmetries and classifications.

Abstract

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.