Pseudo-ovals in even characteristic and ovoidal Laguerre planes

TL;DR

This paper explores the structure of pseudo-ovals in even characteristic projective spaces, establishing their connection to ovoidal Laguerre planes and proving that certain pseudo-ovals are elementary under specific conditions.

Contribution

It demonstrates that pseudo-ovals with all elements inducing Desarguesian spreads are elementary when the field size is even and the dimension is prime, linking geometric structures to algebraic properties.

Findings

Pseudo-ovals of size q^n+1 are characterized as elementary under given conditions.

A connection between dual pseudo-ovals and ovoidal Laguerre planes is established.

Pseudo-hyperovals with all elements inducing Desarguesian spreads are shown to be elementary.

Abstract

Pseudo-arcs are the higher dimensional analogues of arcs in a projective plane: a pseudo-arc is a set $\mathcal{A}$ of $(n-1)$-spaces in $\mathrm{PG}(3n-1,q)$ such that any three span the whole space. Pseudo-arcs of size $q^n+1$ are called pseudo-ovals, while pseudo-arcs of size $q^n+2$ are called pseudo-hyperovals. A pseudo-arc is called elementary if it arises from applying field reduction to an arc in $\mathrm{PG}(2,q^n)$. We explain the connection between dual pseudo-ovals and elation Laguerre planes and show that an elation Laguerre plane is ovoidal if and only if it arises from an elementary dual pseudo-oval. The main theorem of this paper shows that a pseudo-(hyper)oval in $\mathrm{PG}(3n-1,q)$, where $q$ is even and $n$ is prime, such that every element induces a Desarguesian spread, is elementary. As a corollary, we give a characterisation of certain ovoidal Laguerre planes in terms of the derived affine planes.