Regular pseudo-hyperovals and regular pseudo-ovals in even characteristic

TL;DR

This paper provides an alternative proof and improved results on the characterization of regular pseudo-ovals in projective spaces over fields of even characteristic, specifically for PG(3n - 1, q) with q a power of 2.

Contribution

It offers a new proof and stronger results regarding the structure of regular pseudo-ovals in even characteristic projective spaces.

Findings

Alternative proof of regular pseudo-oval characterization

Stronger results on pseudo-oval structure

Applicable to PG(3n - 1, q) with q=2^h, h>1

Abstract

S. Rottey and G. Van de Voorde characterized regular pseudo-ovals of PG(3n - 1, q), q = 2^h, h >1 and n prime. Here an alternative proof is given and slightly stronger results are obtained.

Equations

$$\{{\alpha }_{i0},{\alpha }_{i1},\cdots ,{\alpha }_{i,i-1},{\alpha }_{i,i+1},\cdots ,{\alpha }_{i,q^n+1}\}={\Gamma }_i$$

$$\{{\theta }_1,{\theta }_2,\cdots ,{\theta }_n\}=\{{\theta }_1^{\prime },{\theta }_2^{\prime },\cdots ,{\theta }_n^{\prime }\}.$$

$$\{{\alpha }_{i0},{\alpha }_{i1},\cdots ,{\alpha }_{i,i-1},{\alpha }_{i,i+1},\cdots ,{\alpha }_{i,q^n+1}\}={\Gamma }_i$$

Full text

1. Introduction

Pseudo-ovals and pseudo-hyperovals were introduced in [10]; see also [12]. These objects play a key role in the theory of translation generalized quadrangles [6, 12]. Pseudo-hyperovals only exist in even characteristic. A characterization of regular pseudo-ovals in odd characteristic was given in [2]; see also [12]. In [8] a characterization of regular pseudo-ovals and regular pseudo-hyperovals in $\mathbf{PG}(3n-1,q)$, $q$ even, $q\neq 2$ and $n$ prime, is obtained. Here a shorter proof is given and slightly stronger results are obtained.

2. Ovals and hyperovals

A $k$-arc in $\mathbf{PG}(2,q)$ is a set of $k$ points, $k\geq 3$, no three of which are collinear. Any non-singular conic of $\mathbf{PG}(2,q)$ is a $(q+1)$-arc. If $\mathcal{K}$ is any $k$-arc of $\mathbf{PG}(2,q)$, then $k\leq{q+2}$. For $q$ odd $k\leq q+1$ and for $q$ even a $(q+1)$-arc extends to a $(q+2)$-arc; see [3]. A $(q+1)$-arc is an oval; a $(q+2)$-arc, $q$ even, is a complete oval or hyperoval.

A famous theorem of B. Segre [9] tells us that for $q$ odd every oval of $\mathbf{PG}(2,q)$ is a non-singular conic. For $q$ even, there are many ovals that are not conics [3]; also, there are many hyperovals that do not contain a conic [3].

3. Generalized ovals and hyperovals

Arcs, ovals and hyperovals can be generalized by replacing their points with $m$-dimensional subspaces to obtain generalized $k$-arcs, generalized ovals and generalized hyperovals. These have strong connections to generalized quadrangles, projective planes, circle geometries, flocks and other structures. See [6, 12, 10, 11, 2, 7]. Below, some basic definitions and results are formulated; for an extensive study, many applications and open problems, see [12].

A generalized $k$-arc of $\mathbf{PG}(3n-1,q)$, $n\geq 1$, is a set of $k$ $(n-1)$-dimensional subspaces of $\mathbf{PG}(3n-1,q)$ every three of which generate $\mathbf{PG}(3n-1,q)$. If $q$ is odd then $k\leq q^{n}+1$, if $q$ is even then $k\leq{q^{n}+2}$. Every generalized $(q^{n}+1)$-arc of $\mathbf{PG}(3n-1,q)$, $q$ even, can be extended to a generalized $(q^{n}+2)$-arc.

If $\mathcal{O}$ is a generalized $(q^{n}+1)$-arc in $\mathbf{PG}(3n-1,q)$, then it is a pseudo-oval or generalized oval or [$n-1$]-oval of $\mathbf{PG}(3n-1,q)$. For $n=1$, a [0]-oval is just an oval of $\mathbf{PG}(2,q)$. If $\mathcal{O}$ is a generalized $(q^{n}+2)$-arc in $\mathbf{PG}(3n-1,q)$, $q$ even, then it is a pseudo-hyperoval or generalized hyperoval or [$n-1$]-hyperoval of $\mathbf{PG}(3n-1,q)$. For $n=1$, a [0]-hyperoval is just a hyperoval of $\mathbf{PG}(2,q)$.

If $\mathcal{O}=\{\pi_{0},\pi_{1},\cdots,\pi_{q^{n}}\}$ is a pseudo-oval of $\mathbf{PG}(3n-1,q)$, then $\pi_{i}$ is contained in exactly one $(2n-1)$-dimensional subspace $\tau_{i}$ of $\mathbf{PG}(3n-1,q)$ which has no point in common with $(\pi_{0}\cup\pi_{1}\cup\cdots\cup\pi_{q^{n}})\backslash\pi_{i}$, with $i=0,1,\cdots,q^{n}$; the space $\tau_{i}$ is the tangent space of $\mathcal{O}$ at $\pi_{i}$. For $q$ even the $q^{n}+1$ tangent spaces of $\mathcal{O}$ contain a common $(n-1)$-dimensional space $\pi_{q^{n}+1}$, the nucleus of $\mathcal{O}$; also, $\mathcal{O}\cup\{\pi_{q^{n}+1}\}$ is a pseudo-hyperoval of $\mathbf{PG}(3n-1,q)$. For $q$ odd, the tangent spaces of a pseudo-oval $\mathcal{O}$ are the elements of a pseudo-oval $\mathcal{O^{\ast}}$ in the dual space of $\mathbf{PG}(3n-1,q)$.

4. Regular pseudo-ovals and pseudo-hyperovals

In the extension $\mathbf{PG}(3n-1,q^{n})$ of $\mathbf{PG}(3n-1,q)$, consider $n$ subplanes $\xi_{i}$, $i=1,2,\cdots,n$, that are conjugate in the extension $\mathbb{F}_{q^{n}}$ of $\mathbb{F}_{q}$ and which span $\mathbf{PG}(3n-1,q^{n})$. This means that they form an orbit of the Galois group corresponding to this extension and span $\mathbf{PG}(3n-1,q^{n})$.

In $\xi_{1}$ consider an oval $\mathcal{O}_{1}=\{x_{0}^{(1)},x_{1}^{(1)},\cdots,x_{q^{n}}^{(1)}\}$. Further, let $x_{i}^{(1)},x_{i}^{(2)},\cdots,x_{i}^{(n)}$, with $i=0,1,\cdots,q^{n}$, be conjugate in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$. The points $x_{i}^{(1)},x_{i}^{(2)},\cdots,x_{i}^{(n)}$ define an $(n-1)$-dimensional subspace $\pi_{i}$ over $\mathbb{F}_{q}$ for $i=0,1,\cdots,q^{n}$. Then, $\mathcal{O}=\{\pi_{0},\pi_{1},\cdots,\pi_{q^{n}}\}$ is a generalized oval of $\mathbf{PG}(3n-1,q)$. These objects are the regular or elementary pseudo-ovals. If $\mathcal{O}_{1}$ is replaced by a hyperoval, and so $q$ is even, then the corresponding $\mathcal{O}$ is a regular or elementary pseudo-hyperoval.

All known pseudo-ovals and pseudo-hyperovals are regular.

5. Characterizations

Let $\mathcal{O}=\{\pi_{0},\pi_{1},\cdots,\pi_{q^{n}}\}$ be a pseudo-oval in $\mathbf{PG}(3n-1,q)$. The tangent space of $\mathcal{O}$ at $\pi_{i}$ will be denoted by $\tau_{i}$, with $i=0,1,\cdots,q^{n}$. Choose $\pi_{i}$, $i\in\{0,1,\cdots,q^{n}\}$, and let $\mathbf{PG}(2n-1,q)\subseteq\mathbf{PG}(3n-1,q)$ be skew to $\pi_{i}$. Further, let $\tau_{i}\cap\mathbf{PG}(2n-1,q)=\eta_{i}$ and $\langle\pi_{i},\pi_{j}\rangle\cap\mathbf{PG}(2n-1,q)=\eta_{j}$, with $j\neq i$. Then $\{\eta_{0},\eta_{1},\cdots,\eta_{q^{n}}\}=\Delta_{i}$ is an $(n-1)$-spread of $\mathbf{PG}(2n-1,q)$.

Now, let $q$ be even and let $\pi$ be the nucleus of $\mathcal{O}$. Let $\mathbf{PG}(2n-1,q)\subseteq\mathbf{PG}(3n-1,q)$ be skew to $\pi$. If $\zeta_{j}=\mathbf{PG}(2n-1,q)\cap\langle\pi,\pi_{j}\rangle$, then $\{\zeta_{0},\zeta_{1},\cdots,\zeta_{q^{n}}\}=\Delta$ is an $(n-1)$-spread of $\mathbf{PG}(2n-1,q)$.

Next, let $q$ be odd. Choose $\tau_{i}$, with $i\in\{0,1,\cdots,q^{n}\}$. If $\tau_{i}\cap\tau_{j}=\delta_{j}$, with $j\neq i$, then $\{\delta_{0},\delta_{1},\cdots,\delta_{i-1},\pi_{i},\delta_{i+1},\cdots,\delta_{q^{n}}\}=\Delta^{\star}_{i}$ is an $(n-1)$-spread of $\tau_{i}$.

Finally, let $q$ be even and let $\mathcal{O}=\{\pi_{0},\pi_{1},\cdots,\pi_{q^{n}+1}\}$ be a pseudo-hyperoval in $\mathbf{PG}(3n-1,q)$. Choose $\pi_{i}$, with $i\in\{0,1,\cdots,q^{n}+1\}$, and let $\mathbf{PG}(2n-1,q)\subseteq\mathbf{PG}(3n-1,q)$ be skew to $\pi_{i}$. Let $\langle\pi_{i},\pi_{j}\rangle\cap\mathbf{PG}(2n-1,q)=\eta_{j}$, with $j\neq i$. Then $\{\eta_{0},\eta_{1},\cdots,\eta_{i-1},\eta_{i+1},\cdots,\eta_{q^{n}+1}\}=\Delta_{i}$ is an $(n-1)$- spread of $\mathbf{PG}(2n-1,q)$.

Theorem 5.1** (Casse, Thas and Wild [2]).**

Consider a pseudo-oval $\mathcal{O}$ with $q$ odd. Then at least one of the $(n-1)$-spreads $\Delta_{0},\Delta_{1},\cdots,\Delta_{q^{n}},\Delta^{\star}_{0},\Delta^{\star}_{1},\cdots,\Delta^{\star}_{q^{n}}$ is regular if and only if they all are regular if and only if the pseudo-oval $\mathcal{O}$ is regular. In such a case $\mathcal{O}$ is essentially a conic over $\mathbb{F}_{q^{n}}$.

Theorem 5.2** (Rottey and Van de Voorde [8]).**

Consider a pseudo-oval $\mathcal{O}$ in $\mathbf{PG}(3n-1,q)$ with $q=2^{h}$, $h>1$, $n$ prime. Then $\mathcal{O}$ is regular if and only if all $(n-1)$-spreads $\Delta_{0},\Delta_{1},\cdots,\Delta_{q^{n}}$ are regular.

6. Alternative proof and improvements

Theorem 6.1**.**

Consider a pseudo-hyperoval $\mathcal{O}$ in $\mathbf{PG}(3n-1,q)$, $q=2^{h},h>1$ and $n$ prime. Then $\mathcal{O}$ is regular if and only if all $(n-1)$-spreads $\Delta_{i}$, with $i=0,1,\cdots,q^{n}+1$, are regular.

Proof.  If $\mathcal{O}$ is regular, then clearly all $(n-1)$-spreads $\Delta_{i}$, with $i=0,1,\cdots,q^{n}+1$, are regular.

Conversely, assume that the $(n-1)$-spreads $\Delta_{0},\Delta_{1},\cdots,\Delta_{q^{n}+1}$ are regular. Let $\mathcal{O}=\{\pi_{0},\pi_{1},\cdots,\pi_{q^{n}+1}\}$ and let $\mathcal{\hat{O}}=\{\beta_{0},\beta_{1},\cdots,\beta_{q^{n}+1}\}$ be the dual of $\mathcal{O}$, with $\beta_{i}$ being the dual of $\pi_{i}$.

Choose $\beta_{i},i\in\{0,1,\cdots,q^{n}+1\}$, and let $\beta_{i}\cap\beta_{j}=\alpha_{ij},j\neq i$. Then

[TABLE]

is an $(n-1)$-spread of $\beta_{i}$.

Now consider $\beta_{i},\beta_{j},\Gamma_{i},\Gamma_{j},\alpha_{ij},j\neq i$. In $\Gamma_{j}$ we next consider a $(n-1)$-regulus $\gamma_{j}$ containing $\alpha_{ij}$. The $(n-1)$-regulus $\gamma_{j}$ is a set of maximal spaces of a Segre variety $\mathcal{S}_{1;n-1}$; see Section 4.5 in [4]. The $(n-1)$-regulus $\gamma_{j}$ and the $(n-1)$-spread $\Gamma_{i}$ of $\beta_{i}$ generate a regular $(n-1)$-spread $\Sigma(\gamma_{j},\Gamma_{i})$ of $\mathbf{PG}(3n-1,q)$. This can be seen as follows. The elements of $\Gamma_{i}$ intersect $n$ lines $U_{1},U_{2},\cdots,U_{n}$ which are conjugate in $\mathbb{F}_{q^{n}}$ over $\mathbb{F}_{q}$, that is, they form an orbit of the Galois group corresponding to this extension. Let $\alpha_{ij}\cap U_{l}=\{u_{l}\}$, with $l=1,2,\cdots,n$. Now consider the transversals $T_{1},T_{2},\cdots,T_{n}$ of the elements of $\gamma_{j}$, with $T_{l}$ containing $u_{l}$. The $n$ planes $T_{l}U_{l}=\theta_{l}$ intersect all elements of $\gamma_{j}$ and $\Gamma_{i}$. The $(n-1)$-dimensional subspaces of $\mathbf{PG}(3n-1,q)$ intersecting $\theta_{1},\theta_{2},\cdots,\theta_{n}$ are the elements of the regular $(n-1)$-spread $\Sigma(\gamma_{j},\Gamma_{i})$. The elements of this spread are the points of a plane $\mathbf{PG}(2,q^{n})$, with as lines the $(2n-1)$-dimensional spaces containing at least two (and then $q^{n}+1$) elements of the spread. Hence the $q+2$ elements of $\mathcal{\hat{O}}$ containing an element of $\gamma_{j}$, say $\beta_{i}=\beta_{i_{1}},\beta_{i_{2}},\cdots,\beta_{i_{q+1}},\beta_{i_{q+2}}=\beta_{j}$, are lines of $\mathbf{PG}(2,q^{n})$. Dualizing, the elements $\pi_{i_{1}},\pi_{i_{2}},\cdots,\pi_{i_{q+2}}$ are points of $\mathbf{PG}(2,q^{n})$.

Now consider $\beta_{i_{2}}$ and $\gamma_{j}$, and repeat the argument. Then there arise $n$ planes $\theta^{\prime}_{l}$ intersecting all elements of $\gamma_{j}$ and $\Gamma_{i_{2}}$. The $(n-1)$-dimensional subspaces of $\mathbf{PG}(3n-1,q)$ intersecting $\theta^{\prime}_{1},\theta^{\prime}_{2},\cdots,\theta^{\prime}_{n}$ are the elements of the regular $(n-1)$-spread $\Sigma(\gamma_{j},\Gamma_{i_{2}})$. The elements of this spread are the points of a plane $\mathbf{PG}^{\prime}(2,q^{n})$, with as lines the $(2n-1)$-dimensional spaces containing $q^{n}+1$ elements of the spread. Hence $\beta_{i_{1}},\beta_{i_{2}},\cdots,\beta_{i_{q+2}}$ are lines of $\mathbf{PG}^{\prime}(2,q^{n})$. Dualizing, the elements $\pi_{i_{1}}.\pi_{i_{2}},\cdots,\pi_{i_{q+2}}$ are points of $\mathbf{PG}^{\prime}(2,q^{n})$.

First, assume that $\{\theta_{1},\theta_{2},\cdots,\theta_{n}\}\cap\{\theta^{\prime}_{1},\theta^{\prime}_{2},\cdots,\theta^{\prime}_{n}\}=\emptyset$. Consider $\pi_{i_{1}},\pi_{i_{2}},\pi_{i_{3}},\pi_{i_{4}}$. The planes of $\mathbf{PG}(3n-1,q^{n})$ intersecting these four spaces constitute a set $\mathcal{M}$ of maximal spaces of a Segre variety $\mathcal{S}_{2;n-1}$ [1]. The planes $\theta_{1},\theta_{2},\cdots,\theta_{n},\theta^{\prime}_{1},\theta^{\prime}_{2},\cdots,\theta^{\prime}_{n}$ are elements of $\mathcal{M}$. It follows that $(\theta_{1}\cup\theta_{2}\cup\cdots\cup\theta_{n})\cap(\theta^{\prime}_{1}\cup\theta^{\prime}_{2}\cup\cdots\cup\theta^{\prime}_{n})=\emptyset$.

Consider any $(n-1)$-dimensional subspace $\pi\in\{\pi_{i_{5}},\pi_{i_{6}},\cdots,\pi_{i_{q+2}}\}$ of $\mathbf{PG}(3n-1,q)$. We will show that $\pi$ is a maximal subspace of $\mathcal{S}_{2;n-1}$. Let $\theta_{i}\cap\pi_{j}=\{t_{ij}\},\theta_{i}^{\prime}\cap\pi_{j}=\{t_{ij}^{\prime}\},i=1,2,\cdots,n,j=i_{1},i_{2},\cdots,i_{q+2}$. If $t_{ij_{1}}t_{ij_{2}}\cap t_{ij_{3}}t_{ij_{4}}=\{v_{i}\},t_{ij_{1}}^{\prime}t_{ij_{2}}^{\prime}\cap t_{ij_{3}}^{\prime}t_{ij_{4}}^{\prime}=\{v_{i}^{\prime}\}$, with $j_{1},j_{2},j_{3},j_{4}$ distinct, then $v_{1},v_{2},\cdots,v_{n}$ are conjugate and similarly $v_{1}^{\prime},v_{2}^{\prime},\cdots,v_{n}^{\prime}$ are conjugate. Hence $\langle v_{1},v_{2},\cdots,v_{n}\rangle=\langle v_{1}^{\prime},v_{2}^{\prime},\cdots,v_{n}^{\prime}\rangle$ defines a $(n-1)$-dimensional space over $\mathbb{F}_{q}$ which intersects $\theta_{1},\theta_{2},\cdots,\theta_{n}^{\prime}$ (over $\mathbb{F}_{q^{n}}$).The points $t_{ij}$, with $j=i_{1},i_{2},\cdots,i_{q+2}$, generate a subplane of $\theta_{i}$, and the points $t_{ij}^{\prime}$, with $j=i_{1},i_{2},\cdots,i_{q+2}$, generate a subplane of $\theta_{i}^{\prime}$, with $i=1,2,\cdots,n$. Let $q=2^{h}$ and let $\mathbb{F}_{2^{v}}$ be the subfield of $\mathbb{F}_{q^{n}}=\mathbb{F}_{2^{hn}}$ over which these subplanes are defined; so $v|hn$. Then $v<hn$ as otherwise the spreads of $\mathbf{PG}(3n-1,q)$ defined by $\theta_{1},\theta_{2},\cdots,\theta_{n}$ and $\theta_{1}^{\prime},\theta_{2}^{\prime},\cdots,\theta_{n}^{\prime}$ coincide, clearly not possible. The $(n-1)$-regulus $\gamma_{j}$ implies that the subplanes contain a line over $\mathbb{F}_{q}$, so $h|v$. As $n$ is prime we have $v=h$, so $2^{v}=q$. Hence the $2n$ subplanes are defined over $\mathbb{F}_{q}$. It follows that the $q+2$ elements $\pi_{i_{1}},\pi_{i_{2}},\cdots,\pi_{i_{q+2}}$ are maximal subspaces of the Segre variety $\mathcal{S}_{2;n-1}$. Hence $\pi$ is a maximal subspace of $\mathcal{S}_{2;n-1}$. It follows that $\pi_{1},\pi_{2},\cdots,\pi_{q+2}$ are maximal subspaces of $\mathcal{S}_{2;n-1}$.

Now consider a $\mathbf{PG}(2,q)$ which intersects $\pi_{i_{1}},\pi_{i_{2}},\pi_{i_{3}},\pi_{i_{4}}$. The $(n-1)$-dimensional spaces $\pi_{i_{1}},\pi_{i_{2}},\cdots,\pi_{i_{q+2}}$ are maximal spaces of $\mathcal{S}_{2;n-1}$ which intersect $\mathbf{PG}(2,q)$; they are maximal spaces of the Segre variety $\mathcal{S}_{2;n-1}\cap\mathbf{PG}(3n-1,q)$ of $\mathbf{PG}(3n-1,q)$.

Consider $\pi_{i_{1}}$ and also a $\mathbf{PG}(2n-1,q)$ skew to $\pi_{i_{1}}$. If we project $\pi_{i_{2}},\pi_{i_{3}},\cdots,\pi_{i_{q+2}}$ from $\pi_{i_{1}}$ onto $\mathbf{PG}(2n-1,q)$, then by the foregoing paragraph the $q+1$ projections constitute a $(n-1)$-regulus of $\mathbf{PG}(2n-1,q)$. Similarly, if we project from $\pi_{i_{s}}$, $s$ any element of $\{1,2,\cdots,q+2\}$. Equivalently, if $s\in\{1,2,\cdots,q+2\}$ then the spaces $\beta_{i_{s}}\cap\beta_{i_{t}}$, with $t=1,2,\cdots,s-1,s+1,\cdots,q+2$, form a $(n-1)$-regulus of $\beta_{i_{s}}$.

Now assume that the condition $\{\theta_{1},\theta_{2},\cdots,\theta_{n}\}\cap\{\theta^{\prime}_{1},\theta^{\prime}_{2},\cdots,\theta^{\prime}_{n}\}=\emptyset$ is satisfied for any choice of $\beta_{i},\beta_{j},\gamma_{j},\beta_{i_{2}}$. In such a case every $(n-1)$-regulus contained in a spread $\Gamma_{s}$ defines a Segre variety $\mathcal{S}_{2;n-1}$ over $\mathbb{F}_{q}$. Let us define the following design $\mathcal{D}$. Points of $\mathcal{D}$ are the elements of $\mathcal{\hat{O}}$, a block of $\mathcal{D}$ is a set of $q+2$ elements of $\mathcal{\hat{O}}$, containing at least one space of a $(n-1)$-regulus contained in some regular spread $\Gamma_{s}$, and incidence is containment. Then $\mathcal{D}$ is a $4-(q^{n}+2,q+2,1)$ design. By Kantor [5] this implies that $q=2$, a contradiction.

Consequently, we may assume that for at least one quadruple $\beta_{i},\beta_{j},\gamma_{j},\beta_{i_{2}}$ we have

[TABLE]

In such a case the $q^{n}+2$ elements of $\mathcal{\hat{O}}$ are lines of the plane $\mathbf{PG}(2,q^{n})$. It follows that $\mathcal{O}$ is regular. $\blacksquare$

Theorem 6.2**.**

Consider a pseudo-oval $\mathcal{O}$ in $\mathbf{PG}(3n-1,q)$, with $q=2^{h},h>1$ and $n$ prime. Then $\mathcal{O}$ is regular if and only if all $(n-1)$-spreads $\Delta_{0},\Delta_{1},\cdots,\Delta_{q^{n}}$ are regular.

Proof.  If $\mathcal{O}$ is regular, then clearly all $(n-1)$-spreads $\Delta_{0},\Delta_{1},\cdots,\Delta_{q^{n}}$ are regular.

Conversely, assume that the $(n-1)$-spreads $\Delta_{0},\Delta_{1},\cdots,\Delta_{q^{n}}$ are regular. Let $\mathcal{O}=\{\pi_{0},\pi_{1},\cdots,\pi_{q^{n}}\}$, let $\pi_{q^{n}+1}$ be the nucleus of $\mathcal{O}$, let $\mathcal{\bar{O}}=\mathcal{O}\cup\{\pi_{q^{n}+1}\}$, let $\mathcal{\hat{O}}$ be the dual of $\mathcal{O}$, let $\mathcal{\hat{\bar{O}}}$ be the dual of $\mathcal{\bar{O}}$, and let $\beta_{i}$ be the dual of $\pi_{i}$.

Choose $\beta_{i},i\in\{0,1,\cdots,q^{n}+1\}$, and let $\beta_{i}\cap\beta_{j}=\alpha_{ij},j\neq i$. Then

[TABLE]

is an $(n-1)$-spread of $\beta_{i}$.

Now consider $\beta_{i},\beta_{j},\Gamma_{i},\Gamma_{j},\alpha_{ij}$, with $j\neq i$ and $i,j\in\{0,1,\cdots,q^{n}\}$. In $\Gamma_{j}$ we next consider a $(n-1)$-regulus $\gamma_{j}$ containing $\alpha_{ij}$ and $\alpha_{j,q^{n}+1}$. The $(n-1)$-regulus $\gamma_{j}$ is a set of maximal spaces of a Segre variety $\mathcal{S}_{1;n-1}$. The $(n-1)$-regulus $\gamma_{j}$ and the $(n-1)$-spread $\Gamma_{i}$ of $\beta_{i}$ generate a regular $(n-1)$-spread $\Sigma(\gamma_{j},\Gamma_{i})$ of $\mathbf{PG}(3n-1,q)$. Such as in the proof of Theorem 6.1 we introduce the elements $U_{l},u_{l},T_{l},\theta_{l},l=1,2,\cdots,n$, and the plane $\mathbf{PG}(2,q^{n})$. The $q+2$ elements of $\mathcal{\hat{\bar{O}}}$ containing an element of $\gamma_{j}$, say $\beta_{i}=\beta_{i_{1}},\beta_{i_{2}},\cdots,\beta_{i_{q}},\beta_{j}=\beta_{i_{q+1}},\beta_{q^{n}+1}$, are lines of $\mathbf{PG}(2,q^{n})$. Dualizing, the elements $\pi_{i_{1}},\pi_{i_{2}},\cdots,\pi_{i_{q+1}},\pi_{q^{n}+1}$ are points of $\mathbf{PG}(2,q^{n})$.

Now consider $\beta_{i_{2}}$ and $\gamma_{j}$, and repeat the argument. Then there arise $n$ planes $\theta^{\prime}_{l}$ of $\mathbf{PG}(3n-1,q^{n})$ intersecting all elements of $\gamma_{j}$ and $\Gamma_{i_{2}}$, and a $(n-1)$-spread $\Sigma(\gamma_{j},\Gamma_{i_{2}})$ of $\mathbf{PG}(3n-1,q)$. The elements of this spread are the points of a plane $\mathbf{PG}^{\prime}(2,q^{n})$. The spaces $\beta_{i_{1}},\beta_{i_{2}},\cdots,\beta_{i_{q+1}},\beta_{q^{n}+1}$ are lines of $\mathbf{PG}^{\prime}(2,q^{n})$. Dualizing, the elements $\pi_{i_{1}},\pi_{i_{2}},\cdots,\pi_{i_{q+1}},\pi_{q^{n}+1}$ are points of $\mathbf{PG}^{\prime}(2,q^{n})$.

First, assume that $\{\theta_{1},\theta_{2},\cdots,\theta_{n}\}\cap\{\theta^{\prime}_{1},\theta^{\prime}_{2},\cdots,\theta^{\prime}_{n}\}=\emptyset$. Consider $\pi_{i_{1}},\pi_{i_{2}},\pi_{i_{3}},\pi_{i_{4}}$. The planes of $\mathbf{PG}(3n-1,q^{n})$ intersecting these four spaces constitute a set $\mathcal{M}$ of maximal spaces of a Segre variety $\mathcal{S}_{2;n-1}$. The planes $\theta_{1},\theta_{2},\cdots,\theta_{n},\theta^{\prime}_{1},\theta^{\prime}_{2},\cdots,\theta^{\prime}_{n}$ are elements of $\mathcal{M}$. It follows that $(\theta_{1}\cup\theta_{2}\cup\cdots\cup\theta_{n})\cap(\theta^{\prime}_{1}\cup\theta^{\prime}_{2}\cup\cdots\cup\theta^{\prime}_{n})=\emptyset$. Let $\pi\in\{\pi_{i_{5}},\pi_{i_{6}},\cdots,\pi_{i_{q+1}},\pi_{q^{n}+1}\}$. As in the proof of Theorem 6.1 one shows that $\pi$ is a maximal subspace of $\mathcal{S}_{2;n-1}$. It follows that $\pi_{i_{1}},\pi_{i_{2}},\cdots,\pi_{i_{q+1}},\pi_{q^{n}+1}$ are maximal subspaces of $\mathcal{S}_{2;n-1}$.

Next consider a $\mathbf{PG}(2,q)$ which intersects $\pi_{i_{1}},\pi_{i_{2}},\pi_{i_{3}},\pi_{i_{4}}$. The $(n-1)$-dimensional spaces $\pi_{i_{1}},\pi_{i_{2}},\cdots,\pi_{i_{q+1}},\pi_{q^{n}+1}$ are maximal spaces of $\mathcal{S}_{2;n-1}$ which intersect the plane $\mathbf{PG}(2,q)$; they are maximal spaces of the Segre variety $\mathcal{S}_{2;n-1}\cap\mathbf{PG}(3n-1,q)$ of $\mathbf{PG}(3n-1,q)$. Such as in the proof of Theorem 6.1 it follows that the spaces $\beta_{q^{n}+1}\cap\beta_{i_{t}}$, with $t=1,2,\cdots,q+1$, form a $(n-1)$-regulus of $\beta_{q^{n}+1}$.

Now assume that the condition $\{\theta_{1},\theta_{2},\cdots,\theta_{n}\}\cap\{\theta^{\prime}_{1},\theta^{\prime}_{2},\cdots,\theta^{\prime}_{n}\}=\emptyset$ is satisfied for any choice of $\beta_{i},\beta_{j},\gamma_{j},\beta_{i_{2}}$, $j\neq i$ and $i,j\in\{0,1,\cdots,q^{n}\}$. Let $\alpha_{1},\alpha_{2},\alpha_{3}$ be distinct elements of $\Gamma_{q^{n}+1}$. Then $\beta_{i},\beta_{j},\gamma_{j},\beta_{i_{2}}$ can be chosen in such a way that $\alpha_{1}\in\beta_{i},\alpha_{2}\in\beta_{j},\alpha_{2}\in\gamma_{j},\beta_{i_{2}}\cap\beta_{j}\in\gamma_{j}$ with $\alpha_{3}\in\beta_{i_{2}}$. Hence the $(n-1)$-regulus in $\beta_{q^{n}+1}$ defined by $\alpha_{1},\alpha_{2},\alpha_{3}$ is subset of $\Gamma_{q^{n}+1}$. From [4] now follows that the $(n-1)$-spread $\Gamma_{q^{n}+1}$ of $\beta_{q^{n}+1}$ is regular. By Theorem 6.1 the pseudo-hyperoval $\mathcal{\bar{O}}$ is regular, and so $\mathcal{O}$ is regular. But in such a case the condition $\{\theta_{1},\theta_{2},\cdots,\theta_{n}\}\cap\{\theta^{\prime}_{1},\theta^{\prime}_{2},\cdots,\theta^{\prime}_{n}\}=\emptyset$ is never satisfied, a contradiction.

Consequently, we may assume that for at least one quadruple $\beta_{i},\beta_{j},\gamma_{j},\beta_{i_{2}}$ we have $\{\theta_{1},\theta_{2},\cdots,\theta_{n}\}=\{\theta^{\prime}_{1},\theta^{\prime}_{2},\cdots,\theta^{\prime}_{n}\}$. In such a case the $q^{n}+2$ elements of $\mathcal{\hat{\bar{O}}}$ are lines of the plane $\mathbf{PG}(2,q^{n})$. It follows that $\mathcal{\bar{O}}$, and hence also $\mathcal{O}$, is regular. $\blacksquare$

Theorem 6.3**.**

Consider a pseudo-hyperoval $\mathcal{O}$ in $\mathbf{PG}(3n-1,q)$, $q=2^{h},h>1$ and $n$ prime. Then $\mathcal{O}$ is regular if and only if at least $q^{n}-1$ elements of $\{\Delta_{0},\Delta_{1},\cdots,\Delta_{q^{n}+1}\}$ are regular.

Proof.  If $\mathcal{O}$ is regular, then clearly all $(n-1)$-spreads $\Delta_{i}$, with $i=0,1,\cdots,q^{n}+1$, are regular.

Conversely, assume that $\rho$, with $\rho\geq q^{n}-1$, elements of $\{\Delta_{0},\Delta_{1},\cdots,\Delta_{q^{n}+1}\}$ are regular.

If $\rho=q^{n}+2$, then $\mathcal{O}$ is regular by Theorem 6.1; if $\rho=q^{n}+1$, then $\mathcal{O}$ is regular by Theorem 6.2.

Now assume that $\rho=q^{n}$ and that $\Delta_{2},\Delta_{3},\cdots,\Delta_{q^{n}+1}$ are regular. We have to prove that $\Delta_{0}$ is regular. We use the arguments in the proof of Theorem 6.2. If one of the elements $\alpha_{1},\alpha_{2},\alpha_{3}$, say $\alpha_{1}$, in the proof of Theorem 6.2 is $\beta_{0}\cap\beta_{1}$, then let $\gamma_{j}$ contain $\beta_{j}\cap\beta_{i},\beta_{j}\cap\beta_{0},\beta_{j}\cap\beta_{1}$ and let $\beta_{i_{2}}\neq\beta_{1}$, with $i,j\in\{2,3,\cdots,q^{n}+1\}$. Now see the proof of the preceding theorem.

Finally, assume that $\rho=q^{n}-1$ and that $\Delta_{3},\Delta_{4},\cdots,\Delta_{q^{n}+1}$ are regular. We have to prove that $\Delta_{0}$ is regular. We use the arguments in the proof of Theorem 6.2. If exactly one of the elements $\alpha_{1},\alpha_{2},\alpha_{3}$, say $\alpha_{1}$, in the proof of Theorem 6.2 is $\beta_{0}\cap\beta_{1}$ or $\beta_{0}\cap\beta_{2}$, then proceed as in the preceding paragraph with $\beta_{i_{2}}\neq\beta_{1},\beta_{2}$. Now assume that two of the elements $\alpha_{1},\alpha_{2},\alpha_{3}$, say $\alpha_{1}$ and $\alpha_{2}$, are $\beta_{0}\cap\beta_{1}$ and $\beta_{0}\cap\beta_{2}$. Now consider all $(n-1)$-reguli in $\Delta_{0}$ containing $\alpha_{1}$ and $\alpha_{3}$, and assume, by way of contradiction, that no one of these $(n-1)$-reguli contains $\alpha_{2}$. The number of these $(n-1)$-reguli is $\frac{q^{n}-2}{q-1}$, and so $q=2$, a contradiction. It follows that the $(n-1)$-regulus in $\beta_{0}$ defined by $\alpha_{1},\alpha_{2},\alpha_{3}$ is contained in $\Delta_{0}$. Now we proceed as in the proof of Theorem 6.2. $\blacksquare$

7. Final remarks

7.1. The cases $q=2$ and $n$ not prime

For $q=2$ or $n$ not prime other arguments have to be developed.

7.2. Improvement of Theorem 6.3

Let $\mathcal{D}=(P,B,\in)$ be an incidence structure satisfying the following conditions.

• (i)

$|P|=q^{n}+1$, $q$ even, $q\neq 2$;

• (ii)

the elements of $B$ are subsets of size $q+1$ of $P$ and every three distinct elements of $P$ are contained in at most one element of $B$;

• (iii)

Q is a subset of size $\delta$ of $P$ such that any triple of elements in $P$ with at most one element in $Q$, is contained in exactly one element of $B$;

Assumption : Any such $\mathcal{D}$ is a $3-(q^{n}+1,q+1,1)$ design whenever $\delta\leq\delta_{0}$ with $\delta_{0}\leq q-2$.

Theorem 7.1**.**

Consider a pseudo-hyperoval $\mathcal{O}$ in $\mathbf{PG}(3n-1,q)$, $q=2^{h},h>1$ and $n$ prime. Then $\mathcal{O}$ is regular if and only if at least $q^{n}+1-\delta_{0}$ elements of $\{\Delta_{0},\Delta_{1},\cdots,\Delta_{q^{n}+1}\}$ are regular.

Proof.  Similar to the proof of Theorem 6.3. $\blacksquare$

7.3. Acknowledgement

We thank S. Rottey and G. Van de Voorde for several helpful discussions.