Ovoidal fibrations in $PG(3,q), q$ even

TL;DR

This paper proves a geometric property of ovoid partitions in projective 3-space over even fields, relating lines and their polars to tangent ovoids, using coding theory and symplectic forms.

Contribution

It establishes a new relation between lines, their polars, and tangent ovoids in $PG(3,q)$ for even q, utilizing linear code radicals and symplectic geometry.

Findings

Lines and their polars are tangent to distinct ovoids.

The radical of the associated linear code has codimension 1.

The geometric configuration is characterized for partitions by ovoids.

Abstract

We prove that, given a partition of the point-set of $PG(3,q), q=2^n >2$, by ovoids $\{\theta_i\}^q_{i=0}$ of $PG(3,q)$ and a line $\ell$ of $PG(3,q)$, not tangent to $\theta_0$ if $\ell^\perp$ denotes the polar of $\ell$ relative to the symplectic form on $PG(3,q)$ whose isotropic lines are the tangent lines to $\theta_0$, then $\ell$ and $\ell^\perp$ are tangent to distinct ovoids $\theta_j, \theta_k$, both distinct from $\theta_0$. This uses the fact that the radical of the linear code generated by the dual duals $\ell\cup \ell^\perp$ of the hyperbolic quadrics , with $\ell$ and $\ell^\perp$ as above, is of codimension $1$

Equations

$$Q\left({\Sigma }_{1\le i<j\le 4}{\lambda }_{ij}{e}_i\wedge e_j\right)={\lambda }_{12}{\lambda }_{34}+{\lambda }_{13}{\lambda }_{24}+{\lambda }_{14}{\lambda }_{23}$$

$$p\left(e_1\wedge e_2\right)$$

$$p\left(e_3\wedge e_4\right)$$

$$[{\mathbb{F}}_2^{\mathbb{D}}/Rad({\mathbb{F}}_2^{\mathbb{D}}):{\mathbb{F}}_2]=1$$

$$di{m}_{{\mathbb{F}}_2}(Ho{m}_{{\mathbb{F}}_{2}G}(In{d}_L^G({\mathbb{F}}_2)),{\mathbb{F}}_2)=di{m}_{{\mathbb{F}}_2}(Ho{m}_{{\mathbb{F}}_{2}L}({\mathbb{F}}_2,{\mathbb{F}}_2))=1.$$

$$Ho{m}_{k\Omega (f_{\hslash })}(k,V_I)\cong Ho{m}_{k\Omega (f_{\hslash })}(V_{I_e},V_{I_o})=\{0\}.$$

$$di{m}_k(Ho{m}_{kG}(In{d}_L^G(k),V_I))=di{m}_k(Ho{m}_{kG}(In{d}_{O(f_{\hslash })}^G(k),V_I)).$$

$$di{m}_k(Ho{m}_{kG}(In{d}_L^G(k),V_I))$$

$$=di{m}_k(Ho{m}_{kO(f_{\hslash })}(k,V_I))$$

$$\le di{m}_k(Ho{m}_{k\Omega (f_{\hslash })}(k,V_I))=0$$

$$\sum _{t\in T}t(m+m^{\perp })=E_i+E_i=0.$$

Full text

Keywords: Dual grid; Elliptic quadric; Frobenius reciprocity; Tits Ovoid

MSC (2010): 05B25, 51E22, 94B05

1 Introduction and Statement of the Result

An ovoid in $PG(3,q)$ is a set of $q^{2}+1$ points, no three collinear. Elliptic quadrics and the Suzuki-Tits ovoids ([24], See [13], 16.4 ), which exist, if and only if, $n$ is odd, are the only known ovoids in $PG(3,q)$ and these are the only ovoids if $n\leq 6$ [17]. Classification of ovoids in $PG(3,q)$ is a fundamental problem in Incidence Geometry. We mention in passing that the elliptic quadrics are the only ovoid in $PG(3,q)$ if $q$ is odd (independently due to Barlotti [4] and Panella [18], see [17], Theorem 2.1, p.178). We recall that a general linear complex in $PG(3,q)$ is the set of all absolute lines with respect to a symplectic polarity of $PG(3,q)$ (that is, a polarity for which all points are absolute). If $P$ is the set of all points of $PG(3,q)$ and $L$ is a general linear complex in $PG(3,q)$, then the incidence system $(P,L)$, which we denote by $W(q)$, is a generalized quadrangle of order $(q,q)$ (see [19], p.37). Since the collineation group $PGL(4,q)$ acts by conjugation transitively on the set of all symplectic polarities of $PG(3,q)$, $W(q)$ is uniquely defined, up to a collineation of $PG(3,q)$. An ovoid of $W(q)$ is a set $\mathcal{O}$ of $q^{2}+1$ points, pairwise noncollinear in $W(q)$. Then, any line $l$ of $PG(3,q)$ meets $\mathcal{O}$ in a point if it is a line of $W(q)$ and in zero or two points if it is not a line of $W(q)$ as shown by a simple count. Thus any ovoid of $W(q)$ is an ovoid of $PG(3,q)$. The following observation, due to Segre [21], proves the converse: let $\mathcal{O}$ be an ovoid of $PG(3,q)$, $q$ even, and $W(\mathcal{O})$ be the incidence system whose points are the points of $PG(3,q)$ and lines are the tangent lines of $\mathcal{O}$. For each $x\in\mathcal{O}$, the union of the set of the $q+1$ tangent lines to $\mathcal{O}$ through $x$ is a plane $\pi_{x}$. The correspondence $x\mapsto\pi_{x}$ defines a symplectic polarity (= null polarity) of $PG(3,q)$ [9] whose absolute lines are precisely the lines of $W(\mathcal{O})$. Thus, $W(\mathcal{O})=g(W(q))$ for some $g\in PGL(4,q)$ and the ovoid $\mathcal{O}$ of $PG(3,q)$ is an ovoid of (a unique copy of) $W(q)$. Hence, the classification of ovoids of $PG(3,q)$, $q$ even, is equivalent to the classification of ovoids in $W(q)$.

It is easy to see that any three mutually skew lines in $PG(3,q)$ have exactly $q+1$ transversals (lines meeting each of the three given lines). Such a set of $q+1$ transversals is called a regulus in $PG(3,q)$. The transversals to the lines in any regulus form another regulus, called the opposite of the given regulus. Thus, any three mutually skew lines of $PG(3,q)$ are in a unique regulus. A spread of $PG(3,q)$ is a set of $q^{2}+1$ mutually skew lines. A spread $\mathcal{S}$ is said to be regular (see [9], 5.1) if the unique regulus containing three lines in $\mathcal{S}$ is contained in $\mathcal{S}$.

An ovoidal fibration $\mathcal{F}=\{\theta_{i}\}_{i=0}^{q}$ of $PG(3,q)$ is a set of $q+1$ ovoids partitioning the point set of $PG(3,q)$. We recall that a Singer group in $PGL(4,q)$ is a cyclic group of order $(q^{2}+1)(q+1)$ acting transitively on the point-set of $P(3,q)$. Let $S$ be a Singer group in $PGL(4,q)$, and $T,K$ be its unique subgroups of orders $q^{2}+1$ and $q+1$, respectively. Then, the point $T$-orbits $\{E_{i}\}_{i=0}^{q}$ in $PG(3,q)$ is an ovoidal fibration by elliptic quadrics (see [1], Lemma 2, p.141); also [10], Theorem 3 and [7]); the set $\mathcal{S}$ of common tangent lines to the $E_{i}$’s is a regular spread in $PG(3,q)$; $T$ acts regularly on the set $\mathcal{S}$ and $K$ acts regularly on each member of $\mathcal{S}$  ([12]; see [1], Lemma 2, p.141). More generally, if $\theta$ is an ovoid of $PG(3,q)$ and $\mathcal{S}$ is contained in the general linear complex $\mathcal{L}=\mathcal{L}(\theta)$ consisting of all tangent lines to $\theta$ (see, Segre [21]) and if $K$ is the group of collineations fixing each line in $\mathcal{S}$, then $K$ is a subgroup of order $q+1$ contained in a Singer subgroup of $PGL(4,q)$ (see [12]) and the set $\{x(\theta):x\in K\}$ of ovoids in $PG(3,q)$ is an ovoidal fibration of $PG(3,q)$ ( see [5], Theorem 3.1, p.161). Further, $\theta$ is the only ovoid of the symplectic generalized quadrangle $(P,\mathcal{L})\simeq W(q)$, where $P$ is the point-set of $PG(3,q)$ (see [1], Lemma 2, p.141).

Let $\theta$ be an ovoid of $W(q)$ and ‘$\perp$’ denote the orthogonality relative to the nondengerate symplectic form on $PG(3,q)$ whose isotropic lines are the tangent lines to $\theta$. Since the symplectic polarity defined by $\theta$ maps an external line to $\theta$ to a secant line to $\theta$ and vice-versa, if $m$ is a line of $PG(3,q)$ which is not tangent to $\theta$, then one of the lines $m,m^{\perp}$ shares two points with $\theta$ and the other is disjoint from it ([13], Corollary 1 of Theorem 16.1.8). Consequently, $\theta$ is an ovoid of $PG(3,q)$ also. Thus, the classification of ovoids in $PG(3,q)$ and in $W(q)$ are equivalent problems. We do not know any examples of ovoidal fibration containing a pair of projectively nonequivalent ovoids (see ([6]) for a discussion of the case $q=8$); and of any examples of fibrations consisting of projectively equivalent ovoids, but with no transitive action of a subgroup of $PGL(4,q)$ on its constituent ovoids .

The subset $m\cup m^{\perp}$ of $P$, with $m,m^{\perp}$ as above, is called a dual grid of $W(q)$. With the lines of $W(q)$ incident with it as ‘lines’, $m\cup m^{\perp}$ is a $(1,q)$-subgeneralized quadrangle of $W(q)$. We denote by $\mathbb{D}$ the set of dual grids in $W(q)$.

Proposition 1.1

Let $\{\theta_{i}\}_{i=0}^{q}$ be an ovoidal fibration of $PG(3,q)$ and $\mathcal{L}_{i}$ denote the general linear complex consisting of all tangent lines to $\theta_{i},0\leq i\leq q$. Then, the following hold:

• (i)

The set $\mathcal{S}$ of common tangent lines to the ovoids $\theta_{i}$ is a regular spread in $PG(3,q)$. Further, $\{\mathcal{L}_{i}\}_{i=0}^{q}$ are the only general linear complexes in $PG(3,q)$ containing $\mathcal{S}$; and $\mathcal{L}_{i}\cap\mathcal{L}_{j}=\mathcal{S}$ for  all $0\leq i\neq j\leq q$.

• (ii)

Each line $\ell$ of $PG(3,q)$ not in $\mathcal{S}$ is tangent to a unique ovoid $\theta_{i}$, secant to $q/2$ ovoids $\theta_{j}$, $j\neq i$, and is disjoint from each of the remaining $q/2$ ovoids $\theta_{k}$ of the fibration.

Our object in this note is to prove

Theorem 1.2

Let $\{\theta_{i}\}_{i=0}^{q}$ be an ovoidal fibration of $PG(3,q)$, $W(q)$ be the generalized quadrangle whose line set is the set $\mathcal{L}(\theta_{0})$ of tangent lines to $\theta_{0}$ and $m$ be a line of $PG(3,q)$ not in $\mathcal{L}(\theta_{0})$. Then, $m$ and its perp $m^{\perp}$ in $W(q)$ are tangent to distinct ovoids $\theta_{i}$, each distinct from $\theta_{0}$.

An intrinsic description of the ovoids $\theta_{i}$ the lines $m$ and $m^{\perp}$ are tangent to may be interesting. We would like to view this note as a contribution towards understanding the packings of $PG(3,q)$ by ovoids.

Our proof of Theorem 1.2 uses a property of the binary code we now define. Let $G$ denote the projective symplectic subgroup $PSp(4,q)$  of $PGL(4,q)$ defined by $W(q)$. Let $\mathbb{F}_{2}^{P}$ and $\mathbb{F}_{2}^{\mathbb{D}}$ denote the $\mathbb{F}_{2}G$-permutation modules on $P$ and $\mathbb{D}$, respectively, and $\eta$ denote the $\mathbb{F}_{2}G$-module homomorphism taking $m\cup m^{\perp}\in\mathbb{D}$ to $\sum_{x\in m\cup m^{\perp}}x\in\mathbb{F}_{2}^{P}$. We identify the characteristic function of a subset of $P$, considered as an element of $\mathbb{F}_{2}^{P}$, with the subset itself. Let $\mathcal{C}$ and $\mathcal{D}$ denote the $\mathbb{F}_{2}G$-submodules of $\mathbb{F}_{2}^{P}$ whose generators are, respectively, the lines of $W(q)$ and the dual grids of $W(q)$. Then, $\eta(\mathbb{F}_{2}^{\mathbb{D}})=\mathcal{D}$. Since any line of $W(q)$ meets a dual grid in zero or two points, $\mathcal{D}$ is contained in the dual code $\mathcal{C}^{\perp}$ of $\mathcal{C}$. Further, it is generated by words of $\mathcal{C}^{\perp}$ of minimum weight ([2], Theorem 1.4 ).

Corollary 1.3

$\mathcal{D}\underset{+}{\subset}\mathcal{C}^{\perp}$.

We need the following

Theorem 1.4

The $\mathbb{F}_{2}G$-radical $U$ of $\mathcal{D}$ is of codimension one in $\mathcal{D}$. Consequently, the sum of two dual grids of $W(q)$ is in the radical $U$.

Lemma 1.5

The group $G$ contains a unique conjugacy class of subgroups $T$ of order $q^{2}+1$ and $T$ is cyclic. Let $\{E_{i}\}_{i=0}^{q}$ be the point $T$-orbits in $PG(3,q)$ and $\mathcal{S}$ denote the set of all common tangent lines to $E_{i}$’s. Let $\ell$ be a line of $PG(3,q)$. Then, $\sum\limits_{t\in T}t(\ell)\in\mathbb{F}_{2}^{P}$ is $P$ (i.e., the ‘all-one’ vector) or the unique ovoid $E_{i}$ the line $\ell$ is tangent to, according as $\ell\in\mathcal{S}$ or $\ell\not\in\mathcal{S}$.

2 Preliminaries

Let $V(q)$ be the vector space of dimension four over $\mathbb{F}_{q}$; $\hbar$ be a non-degenerate symplectic bilinear form on it; and $W(q)$ be the incidence system with the set $P=PG(3,q)$ of all one dimensional subspaces of $V(q)$ as its point-set, the set $L$ of all two dimensional subspaces of $V(q)$ which are isotropic with respect to $\hbar$ as its line-set and symmetrized inclusion as the incidence. Then, $W(q)$ is a regular generalized quadrangle of order $q$ ([19], p.37) and the symplectic group $G$ defined by $\hbar$ acts as incidence preserving permutations on the sets $P$ and $L$.

Let $k$ be an algebraically closed extension field of $\mathbb{F}_{q}$. Let $N=\{0,1,\cdots,2n-1\}$. Addition in $N$ is always taken modulo $2n$. Let $\mathcal{N}$ denote the set of all subsets $I$ of $N$ containing no consecutive elements. Let $V=V\left(q\right)\otimes k$. The natural extension of the symplectic form $\hbar$ to $V$ defined above is also denoted by $\hbar$. Then, $G$ is the subgroup of the algebraic group $Sp(V)\simeq Sp(4,k)$ fixed by the $n$-th power of the Frobenius map $\sigma$ (which is the algebraic group endomorphism of $GL(4,k)$ raising each entry of a matrix to its $2^{nd}$-power). It is well known that $Sp(V)$ has an algebraic group endomorphism $\tau$ with $\tau^{2}=\sigma$ ([23], Theorem 28, p.146). For any non-negative integer $i$, we denote by $V_{i}$ the $Sp(V)$- module whose $k$- vector space structure is the same as that of $V$ and an element $g$ of $Sp(V)$ acts on $V_{i}$ as $\tau^{i}(g)$ would act on $V$. For $I\subseteq N$,  let $V_{I}$ denote the $kG$- module ${\otimes_{i\in I}V}_{i}$ (with $V_{\emptyset}=k$). Then, by Steinberg’s tensor product Theorem ([22], §11), $\{V_{I}:I\subseteq N\}$ is a complete set of inequivalent simple $kG$ -modules. For a $kG$-module $M$, we denote by $[M:V_{I}]$ the multiplicity of $V_{I}$ in a composition series of $M$. We denote by rad$(M)$ the radical of $M$ (that is, the smallest submodule of $M$ with semisimple quotient). We refer to $M/rad(M)$ as the head of $M$.

We now describe a graph automorphism $\tau$ of $Sp(V),$ following ([11], pp.58-60). (The argument presented in loc. cit. constructs a graph automorphism for $G=Sp(V(q))$, however the arguments are valid for $Sp(V)$ also.) Let $\{e_{1},e_{2},e_{3},e_{4}\}$ be an ordered basis of $V$ and $Q$ denote the nondegenerate quadratic form on the exterior square $\Lambda^{2}V$ of $V$ defined by

[TABLE]

(whose zero set in $P(\Lambda^{2}V)$ is the well-known Klein quadric). Let $\beta$ denote the polarization of $Q$ and $\gamma=e_{1}\wedge e_{4}+e_{2}\wedge e_{3}\in\Lambda^{2}V$. Then, the restriction of $Q$ to the hyperplane $U=\{x\in\Lambda^{2}V:\beta\left(x,\gamma\right)=0\}$ of $\Lambda^{2}V$ is a nondegenerate quadratic form; and the restriction of $\beta$ to $U$ is an alternating form with radical $k\gamma.$ The alternating form $\overline{\beta}$ induced by $\beta$ on $\overline{U}=\frac{U}{k\gamma}$ is nondegenerate. So the symplectic space $\left(\overline{U},\overline{\beta}\right)$ is isometric to $\left(V,\hbar\right).$ Let $\overline{p}:\overline{U}\rightarrow V$ be the isometric isomorphism induced by the linear map $p:U\rightarrow V$ defined by

[TABLE]

Then the map taking $g\in Sp(V)$ to $\overline{p}(\overline{\wedge^{2}(g))\text{ }}\overline{p}^{-1}\in Sp(V)$ is a graph automorphism $\tau$ of $Sp(V)$ which, on restriction to $G$, gives a graph automorphism of $G$.

We use the following results:

Lemma 2.1

([8], Corollary 7.11, p.148) Let $K$ be a field of characteristic $p>0$, let $E$ be a field extension of $K$ and let $X$ be a finite group. Then, for each simple $KX$ -module $M,M^{E}=M\otimes_{K}E$ is a direct sum of simple $EX$ -modules, no two of which are isomorphic.

Lemma 2.2

([20], Corollary 6) Let $K,K^{\prime}\in\mathcal{N}$ be distinct and $f_{\hbar}$ be a nondegenerate quadratic form on $V(q)$ of index $2$ which polarizes to $\hbar$. Then, $V_{K}$ and $V_{K^{\prime}}$ are semi-simple $k\Omega(f_{\hbar})$-module with no irreducible factors in common.

3 Proofs

Proof of Theorem 1.4: Let $g\in\mathbb{D}$ and $L$ = Stab${}_{G}(g)$. We view ${\mathbb{F}}_{2}^{\mathbb{D}}$ as the induced module of the trivial $\mathbb{F}_{2}G$- module $\mathbb{F}_{2}$, and write Ind${}_{L}^{G}(\mathbb{F}_{2})={\mathbb{F}}_{2}^{\mathbb{D}}$. Then,

[TABLE]

because, by Frobenius reciprocity ([16], p.689),

[TABLE]

This shows that the trivial module $\mathbb{F}_{2}$ is a summand of the head (that is, the largest semi-simple quotient) of the $\mathbb{F}_{2}G$-module ${\mathbb{F}}_{2}^{\mathbb{D}}$. Let $\mathcal{H}$ denote the set of all hyperbolic quadrics of $W(q)$ and $f_{\hbar}$ be a nondegenerate quadratic form on $V(q)$ of index $2$ which polarizes to $\hbar$. Then, the variety defined by $f_{\hbar}$ is a member of $\mathcal{H}$. Let $O(f_{\hbar})\subseteq G$ denote the orthogonal group of $f_{\hbar}$ and $\Omega(f_{\hbar})$ denote its commutator subgroup. Let $I$ be a nonempty subset of $N$. Then by Lemma 2.2,

[TABLE]

This proves that $V_{I}$ has no nonzero fixed points for the action of $O(f_{\hbar})$.

Since $G$ acts transitively on $\mathcal{H}$ and since the automorphism $\tau$ maps the stabilizer of a hyperbolic quadric in $W(q)$ to the stabilizer of a dual grid in $W(q)$ and vice-versa,

[TABLE]

Again using Frobenius reciprocity, we have

[TABLE]

This proves that the head of the $kG$-module $k^{\mathbb{D}}\cong Ind_{L}^{G}(k)$ is $k$. Now the Lemma 2.1 completes the proof of Theorem 1.4.

Proof of Proposition 1.1: The statement (i) follows from ([3], Theorem 2.2) and ([5], Lemma 2.4 ). Since $|\mathcal{L}_{i}|=(q^{2}+1)(q+1)$ and $\mathcal{L}_{i}\cap\mathcal{L}_{j}=\mathcal{S}$ which has $q^{2}+1$ elements for all $i,j,$ $i\neq j$ (see [3], Corollary 3.3), each line $\ell$ of $PG(3,q)$ not in $\mathcal{S}$ is in $\mathcal{L}_{i}$ for a unique $i$ and $|\ell\cap\theta_{j}|\in\{0,2\}$ for all $j\neq i$. Since $\{\theta_{i}\}$ partitions $PG(3,q)$, (ii) follows.

Proof of Lemma 1.5: The first statement follows from ([11], Lemma 3); the second from ([1], Lemma 2, p.141); and the third follows by Proposition 1.1(ii) and the regular action of $T$ on each $E_{i}$ (see [1], Lemma 4, p.142).

Proof of Theorem 1.2: Let $\mathcal{S}$ be the set of common tangent lines to the $\theta_{i}$’s. Then, $\mathcal{S}$ is a regular spread in $PG(3,q)$ by Proposition 1.1 (i). Let $H$ be the subgroup of $PGL(4,q)$ fixing each line in $\mathcal{S}$. Then $H$ is contained in a Singer subgroup $K$ of $PGL(4,q)$ (see [12]). Consider the unique subgroup $T$ of $K$ of order $q^{2}+1$. The point-orbits of $T$ are elliptic ovoids $E_{i}$, forming an ovoidal fibration of $PG(3,q)$ and $\mathcal{S}$ is also the set of common tangents to $E_{i}$’s (See [1], Lemma 2, p.141). Let $\mathcal{L}_{i}$ be the set of all tangent lines to $\theta_{i}$ and $W(q)=W(\mathcal{L}_{0})$. Since $\mathcal{S}$ is contained in exactly $q+1$ general linear complexes (see Proposition 1.1 (i)), we may assume that $\mathcal{L}_{i}$ is also the set of all tangent lines to $E_{i}$. Thus $\theta_{0}$ and $E_{0}$ are ovoids of $W(q)=W(L_{0})$ and $T$ is a subgroup of $G$ (= $Sp(4,q)$ defined by $W(q)$). Now, assume that $m\cup m^{\perp}$ is a dual grid of $W(q)$ such that $m$ amd $m^{\perp}$ are tangents to $\theta_{i}$ for some $i\neq 0$. Then, both are tangents to $E_{i}$ also. Since $m,m^{\perp}\notin\mathcal{S}$, by Lemma 1.5,

[TABLE]

On the other hand, since for each $t\in T$, $t(m+m^{\perp})\in\mathcal{D}$ and the codimension of $U$ in $\mathcal{D}$ is one, $\sum_{t\in T}t(m+m^{\perp})=m+m^{\perp}$ (mod $U$). Since $G$ is transitive on $\mathbb{D}$, $U$ can not contain any dual grid of W(q). This completes the proof of the theorem.

**Proof of Corollary 1.3: **Let $T$ be a cyclic subgroup of order $q^{2}+1$ contained in the stabilizer in $G$ of an elliptic ovoid $\theta_{0}$ of $W(q)$. Let $\{\theta_{i}\}_{i=0}^{q}$ be the point $T$-orbits in $PG(3,q)$ and $\mathcal{S}$ denote the set of all common tangent lines to $\theta_{i}$’s (See [1], Lemma 2, p.141). Consider the $\mathbb{F}_{2}$-linear map $\sigma:\mathbb{F}_{2}^{P}\longrightarrow\mathbb{F}_{2}^{P}$ defined by $\sigma(w)=\sum\limits_{t\in T}t(w)$. As we noted earlier, $\mathcal{D}\subseteq\mathcal{C}^{\perp}$. Assume that $\mathcal{C}^{\perp}=\mathcal{D}$. Then $\mathcal{C}=(\mathcal{C}^{\perp})^{\perp}=\mathcal{D}^{\perp}\supseteq\mathcal{D}$, as any two dual grids of $W(q)$ intersect in zero or two points. Now, $\sigma(\mathcal{C})=\{\emptyset,P,\theta_{0},P\setminus\theta_{0}\}$ (see Lemma 1.5); however if $m\cup m^{\perp}$ is a dual grid of $W(q)$, then by Lemma 1.5 and Theorem 1.2 $\sigma(m+m^{\perp})=\theta_{i}+\theta_{j}$ for distinct $i$ and $j$ ($i>0,j>0$), a contradiction.