This project investigates orbits of subspaces under classical group actions in Galois geometry, focusing on their interaction with algebraic varieties over finite fields. While the actions of classical groups on low-dimensional spaces are well understood, their behaviour in higher dimensions remains largely unexplored. The study of non-standard group actions on Segre and Veronese varieties provides new insights into the structure of tensor decompositions, secant varieties, and geometric invariants relevant to combinatorics and algebraic geometry.
A key objective is to examine the orbit structure of subspaces in the ambient space of these varieties, with particular emphasis on the classification of planes in the ambient space of the Veronese surface and their connection to nets of quadrics. Understanding these orbits has implications for tensor rank problems and secant varieties, contributing to the broader study of algebraic geometry over finite fields.