Partial ovoids and partial spreads of classical finite polar spaces

We survey the main results on ovoids and spreads, large maximal partial ovoids and large maximal partial spreads, and on small maximal partial ovoids and small maximal partial spreads in classical finite polar spaces. We also discuss the main results on the spectrum problem on maximal partial ovoids and maximal partial spreads in classical finite polar spaces. 1 Classical finite polar spaces The classical finite polar spaces play an important role in incidence geometry. They consist of the non-singular quadrics, the non-singular hermitian varieties, and the symplectic spaces in projective spaces of odd dimension (see [7] and [27] for an introduction to polar spaces). The interest in these incidence structures follows first of all from the fact that they are classical geometrical objects. The study of substructures contained in these classical finite polar spaces also contributes to their geometrical importance. The substructures involved include (partial) ovoids and (partial) spreads. We survey the main results on these (partial) ovoids and (partial) spreads. The focus is first of all put on the known (non-)existence results on ovoids and spreads in classical finite polar spaces. Then the attention is drawn to the main results on large partial ovoids and large partial spreads. This is then followed by focusing on small maximal partial ovoids and small maximal partial spreads. After the discussion of these results, attention is paid to the spectrum problem on maximal partial ovoids and maximal partial spreads. The work on this survey article initiated while the first author was visiting the JustusLiebig-Universität Gießen, Germany. The first author thanks the Fund for Scientific Research Flanders (Belgium) for a research grant. The work on this survey article initiated while the fourth author was visiting the JustusLiebig-Universität Gießen, Germany, with an Alexander von Humboldt Fellowship. The fourth author wishes to thank the Alexander von Humboldt Foundation for granting him this Fellowship.