Exending pseudo-arcs in odd characteristic

TL;DR

This paper explores the structure of pseudo-arcs in odd characteristic projective spaces, establishing conditions under which they are contained in pseudo-conics, and connecting these structures to generalized quadrangles and other geometric configurations.

Contribution

It proves that large pseudo-arcs with certain extension properties are contained in pseudo-conics, extending previous results and linking pseudo-arcs to classical geometric structures.

Findings

Pseudo-arcs larger than the second largest complete arc are contained in pseudo-conics.

Extension of partial spreads to Desarguesian spreads implies containment in pseudo-conics.

Connects pseudo-arcs with generalized quadrangles and classical geometries.

Abstract

A {\em pseudo-arc} in $\mathrm{PG}(3n-1,q)$ is a set of $(n-1)$-spaces such that any three of them span the whole space. A pseudo-arc of size $q^n+1$ is a {\em pseudo-oval}. If a pseudo-oval $\mathcal{O}$ is obtained by applying field reduction to a conic in $\mathrm{PG}(2,q^n)$, then $\mathcal{O}$ is called a {\em pseudo-conic}. We first explain the connection of (pseudo-)arcs with Laguerre planes, orthogonal arrays and generalised quadrangles. In particular, we prove that the Ahrens-Szekeres GQ is obtained from a $q$-arc in $\mathrm{PG}(2,q)$ and we extend this construction to that of a GQ of order $(q^n-1,q^n+1)$ from a pseudo-arc of $\mathrm{PG}(3n-1,q)$ of size $q^n$. The main theorem of this paper shows that if $\mathcal{K}$ is a pseudo-arc in $\mathrm{PG}(3n-1,q)$, $q$ odd, of size larger than the size of the second largest complete arc in $\mathrm{PG}(2,q^n)$, where for one element $K_i$ of $\mathcal{K}$, the partial spread $\mathcal{S}=\{K_1,\ldots,K_{i-1},K_{i+1},\ldots,K_{s}\}/K_i$ extends to a Desarguesian spread of $\mathrm{PG}(2n-1,q)$, then $\mathcal{K}$ is contained in a pseudo-conic. The main result of \cite{Casse} also follows from this theorem.