Regularity conditions for spreads associated with pseudo-ovals

Determine the structure and classification of pseudo-ovals in PG(3n − 1, q) under each of the four unresolved spread-regularity regimes: q = 2, composite n, failure of regularity for all associated spreads, and regularity of at least one associated spread.

Background

A pseudo-oval determines several (n − 1)-spreads, including the spreads Δ_i associated with its elements. For q = 2h with h > 1 and n prime, the paper gives a characterization: all Δ_i are regular exactly when the pseudo-oval is regular. It then lists four cases not settled by that theorem, including non-prime n and weaker assumptions on the number of regular spreads.

References

Open problem 5. From this theorem, the following questions arise. I. What happens when q = 2? II. What happens when n is not prime? III. What happens when not all spreads ∆i are regular? IV. What happens when at least one of the spreads ∆i is regular?

— Arcs, Caps and Generalisations in a Finite Projective Space  (2503.06243 - Hirschfeld et al., 8 Mar 2025) in Open problem 5, Section 7.1, page 7