On Some $2$-Tight Sets of Polar Spaces

Abstract

Let $\mathrm{\Pi }$ be a finite polar space of rank $n\ge 2$ fully embedded into a projective space $\mathrm{\Sigma }$. In this note, we determine all tight sets of $\mathrm{\Pi }$ of the form $({\mathrm{\Sigma }}_1\cap \mathcal{P})\cup ({\mathrm{\Sigma }}_2\cap \mathcal{P})$, where $\mathcal{P}$ denotes the point set of $\mathrm{\Pi }$ and ${\mathrm{\Sigma }}_1,{\mathrm{\Sigma }}_2$ are two mutually disjoint subspaces of $\mathrm{\Sigma }$. In this way, we find two families of $2$-tight sets of elliptic polar spaces that were not described before in the literature.