Geometries arising from trilinear forms on low-dimensional vector spaces

TL;DR

This paper classifies geometries of poles from hyperplanes in Grassmannians related to low-dimensional trilinear forms, revealing new structures and extending known results on line spreads in projective spaces.

Contribution

It characterizes possible pole geometries for 3-forms in dimensions up to 7 and introduces new constructions, extending previous classifications and results.

Findings

Classified geometries of poles for k=3, n≤7.

Proposed new geometric constructions from hyperplanes.

Extended results on line spreads in PG(5, K).

Abstract

Let ${\mathcal G}_k(V)$ be the $k$-Grassmannian of a vector space $V$ with $\dim V=n$. Given a hyperplane $H$ of ${\mathcal G}_k(V)$, we define in [I. Cardinali, L. Giuzzi, A. Pasini, A geometric approach to alternating $k$-linear forms, J. Algebraic Combin. doi:10.1007/s10801-016-0730-6] a point-line subgeometry of ${\mathrm{PG}}(V)$ called the {\it geometry of poles of $H$}. In the present paper, exploiting the classification of alternating trilinear forms in low dimension, we characterize the possible geometries of poles arising for $k=3$ and $n\leq 7$ and propose some new constructions. We also extend a result of [J.Draisma, R. Shaw, Singular lines of trilinear forms, Linear Algebra Appl. doi:10.1016/j.laa.2010.03.040] regarding the existence of line spreads of ${\mathrm{PG}}(5,{\mathbb K})$ arising from hyperplanes of ${\mathcal G}_3(V).$

Tables

Table 1. Table 1: Types for linear functionals on ⋀ 3 V superscript 3 𝑉 \bigwedge^{3}V .

Table 2. Table 2: Matrices associated to forms of rank up to 6 6 6 .

Table 3. Table 3: Matrices associated to forms of rank 7 7 7 .

Table 4. Table 4: Description of the geometries of poles associated to forms of rank up to 6 6 6 .

Table 5. Table 5: Description of the geometry of poles associated to forms of rank 7 7 7 .

Equations

$${\chi }^{\prime }(x_1,\dots ,x_k)=\chi (g(x_1),\dots ,g(x_k)),\forall x_1,\dots ,x_k\in V$$

$$h_u:V/⟨u⟩\times V/⟨u⟩\to \mathbb{K},h_u(x+⟨u⟩,y+⟨u⟩):=h(u\wedge x\wedge y).$$

$${\chi }_u:V\times V\to \mathbb{K},{\chi }_u(x,y)=h(u\wedge x\wedge y)=x^T{M}_{u}y$$

$$(x^T\cdot C_1)u_1+(x^T\cdot C_2)u_2+\cdots +(x^T\cdot C_i)u_i+\cdots +(x^T\cdot C_n)u_n=0,$$

$$(x^T\cdot C_1)\frac{u_1}{u_i}+(x^T\cdot C_2)\frac{u_2}{u_i}+\cdots +x^T\cdot C_i+\cdots +(x^T\cdot C_n)\frac{u_n}{u_i}=0,$$

$$x^T\cdot (\frac{u_1}{u_i}{C}_1+\cdots +C_i+\cdots +\frac{u_n}{u_i}{C}_n)=0.$$

$$C_{i}=-\sum_{\tiny{\begin{array}[]{l}j=1\\ j\not=i\end{array}}}^{n}\frac{u_{j}}{u_{i}}C_{j}.$$

$${\tilde{h}}_{xy}:V\to \mathbb{K},{\tilde{h}}_{xy}(u)=h(u\wedge x\wedge y)$$

$$\det M_u^{(i)}=(d_i(u_1,\dots ,u_n){)}^2.$$

$$d_i(u_1,\dots ,u_n)=u_i^{{\alpha }_i}{g}_i(u_1,\dots ,u_n)$$

$$d_i(u_1,\dots ,u_{i-1},0,u_{i+1},\dots ,u_n)=0$$

$$C_{j}=-\sum_{\tiny{\begin{array}[]{l}k=1\\ k\not=j\end{array}}}^{n}\frac{u_{k}}{u_{j}}C_{k}.$$

$$d_i(u_1,\dots ,u_n)=u_i{d}_i^{\prime }(u_1,\dots ,u_n),$$

$$P(H)=C(P(H)\setminus {\Pi }_i)=C({\overline{\Gamma }}_i\setminus {\Pi }_i),$$

$$f(u_1,\dots ,u_n):=u_{n_1}\cdot f_1(u_1,\dots ,u_{n_1})f_2(u_{n_1},\dots ,u_n).$$

$$h_i(x,y,z)={\overline{h}}_i({\pi }_i(x),{\pi }_i(y),{\pi }_i(z))$$

$$M_u=\left(\begin{array}{ccc}{M}_1& (v_{n_1}^1)& 0\\ -{(v_{n_1}^1)}^T& 0& -{(v_{n_1}^2)}^T\\ 0& (v_{n_1}^2)& M_2\end{array}\right)$$

$$0=\det M_u^{(n_1)}=\det M_1\cdot \det M_2=\det {\bar{M}}_1^{(n_1)}\cdot \det {\bar{M}}_2^{(n_1)}.$$

$$P(H)\setminus {\Pi }_{n_1}=\Delta \setminus {\Pi }_{n_1}$$

$$P(H)=(P(H)\setminus {\Pi }_{n_1})\cup {\Pi }_{n_1}=(\Delta \setminus {\Pi }_{n_1})\cup {\Pi }_{n_1}$$

$$g(u_1,\dots ,u_{n+2}):=g_1(u_1,\dots ,u_n)\cdot u_n=\left(\prod _{i=1}^{(n-3)/2}{u}_{2i+1}\right)\cdot u_n=\prod _{i=1}^{(n-1)/2}{u}_{2i+1}.$$

$$\left.\begin{array}[]{rcll}\chi(x_{1},\dots,x_{k})&=&0&\mbox{if}~{}x_{i}\in V_{1}~{}\mbox{for some}~{}1\leq i\leq k,\\ \chi(x_{1},\dots,x_{k})&=&\chi_{0}(x_{1},\dots,x_{k})&\mbox{if}~{}x_{i}\in V_{0}~{}\mbox{for all}~{}1\leq i\leq k,\end{array}\right\}$$

$$H=\{X\in G_k(V):X\cap V_1\ne 0\}.$$

$$\mathrm{Exp}(H_0):=\{X\in {\mathcal{G}}_k(V):\text{mbox}eitherX\subset V_0\text{mbox}orX\cap V_0\in H_0\}.$$

$$h(e_{i_{1}}\wedge\cdots\wedge e_{i_{k}})=\left\{\begin{array}[]{l}0~{}\mbox{if}~{}i_{k}<n,\\ h_{0}(e_{i_{1}}\wedge\cdots\wedge e_{i_{k-1}})~{}\mbox{if}~{}i_{k}=n.\end{array}\right.$$

$$h_i(x\wedge y\wedge z)={\bar{h}}_i(x_i\wedge y_i\wedge z_i)$$

$$h((x_0+x_1)\wedge (y_0+y_1)\wedge (z_0+z_1))={\bar{h}}_0(x_0\wedge y_0\wedge z_0)+{\bar{h}}_1(x_1\wedge y_1\wedge z_1).$$

$${\epsilon }_2(R^{\uparrow }(H))=[{\epsilon }_2(R^{\uparrow }(H_0))+{\epsilon }_2(R^{\uparrow }(H_1))+V_0\wedge V_1]\cap {\epsilon }_2({\mathcal{G}}_2)$$

$$M_u=\left(\begin{array}{cc}{M}_{u_0}& 0\\ 0& M_{u_1}\end{array}\right)$$

$${\chi }_u(x,y)=x^T{M}_{u}y=x_0^T{M}_{u_0}{y}_0+x_1^T{M}_{u_1}{y}_1.$$

Full text

Keywords: Grassmann Geometry; Hyperplanes; Multilinear forms.

MSC: 15A75; 14M15; 15A69.

1 Introduction

Denote by $V$ a $n$-dimensional vector space over a field $\mathbb{K}$. For any fixed $1\leq k<n$ the $k$-Grassmannian ${\mathcal{G}}_{k}(V)$ of $V$ is the point-line geometry whose points are the $k$–dimensional vector subspaces of $V$ and whose lines are the sets $\ell_{Y,Z}:=\{X~{}\colon~{}Y\subset X\subset Z,~{}\dim X=k\}$ where $Y$ and $Z$ are subspaces of $V$ with $Y\subset Z,$ $\dim Y=k-1$ and $\dim Z=k+1$. Incidence is containment.

It is well known that the geometry $\mathcal{G}_{k}(V)$ affords a full projective embedding, called Grassmann (or Plücker) embedding and denoted by $\varepsilon_{k}$, sending every $k$-subspace $\langle v_{1},\ldots,v_{k}\rangle$ of $V$ to the point $[v_{1}\wedge\cdots\wedge v_{k}]$ of $\mathrm{PG}(\bigwedge^{k}V)$, where we adopt the notation $[u]$ to refer to the projective point represented by the vector $u$. Also, for $X\subseteq V$ we put $[X]:=\{[x]:x\in\langle X\rangle\}$ for the points of the projective space induced by $\langle X\rangle$.

A hyperplane $H$ of $\mathcal{G}_{k}(V)$ is a proper subspace of $\mathcal{G}_{k}(V)$ such that any line of $\mathcal{G}_{k}(V)$ is either contained in $H$ or it intersects $H$ in just one point.

It is well known, see Shult [shult92] and also Havlicek [H81], Havlicek and Zanella [HZ08] and De Bruyn [B09], that the hyperplanes of $\mathcal{G}_{k}(V)$ all arise from the Plücker embedding of $\mathcal{G}_{k}(V)$ in $\mathrm{PG}(\bigwedge^{k}V)$, i.e. they bijectively correspond to proportionality classes of non-zero linear functionals on $\bigwedge^{k}V$. More in detail, for any hyperplane $H$ of $\mathcal{G}_{k}(V)$ there is a non-null linear functional $h$ on $\bigwedge^{k}V$ such that $H=\varepsilon_{k}^{-1}([\ker(h)]\cap\varepsilon_{k}(\mathcal{G}_{k}(V)))$. Equivalently, if $\chi_{h}:V\times\ldots\times V\rightarrow\mathbb{K}$ is the alternating $k$-linear form on $V$ associated to the linear functional $h$, defined by the clause $\chi_{h}(v_{1},\ldots,v_{k}):=h(v_{1}\wedge\cdots\wedge v_{k})$, then the hyperplane $H$ is the set of the $k$-subspaces of $V$ where $\chi_{h}$ identically vanishes. So, the hyperplanes of $\mathcal{G}_{k}(V)$ bijectively correspond also to proportionality classes of non-trivial alternating $k$-linear forms of $V$.

In a recent paper [ILP17] we introduced the notion of $i$-radical of a hyperplane $H$ of $\mathcal{G}_{k}(V)$. In the present work we shall just consider the case of the lower radical $R_{\downarrow}(H)$, for $i=1$, and that of the upper radical $R^{\uparrow}(H)$, for $i=k-1$. The lower radical $R_{\downarrow}(H)$ of $H$ is the set of points $[p]\in\mathrm{PG}(V)$ such that all $k$-spaces $X$ with $p\in X$ belong to $H$; the upper radical $R^{\uparrow}(H)$ of $H$ is the set of $(k-1)$-subspaces $Y$ of $V$ such that all $k$-spaces through $Y$ belong to $H$.

In the same paper [ILP17] we investigated the problem of determining under which conditions the upper radical of a given hyperplane might be empty. Working in the case $k=3$, we also defined a point-line subgeometry $\mathcal{P}(H)=(P(H),R^{\uparrow}(H))$ of $\mathrm{PG}(V)$ called the geometry of poles of $H$, whose points are called $H$-poles, a point $[p]\in P(H)$ and a line $\ell\in R^{\uparrow}(H)$ being incident when $p\in\ell$ (see Section 1.1). Some aspects of this geometry had already been studied, under slightly different settings, in [DS10, DS14]. It has been shown in [DS10] that the set $P(H)$ is actually either $\mathrm{PG}(V)$ or an algebraic hypersurface in $\mathrm{PG}(V)$; for more details see Section 1.1.

In this paper we shall focus on the geometry $\mathcal{P}(H)$, providing explicit equations for its points and lines and also a geometric description in the cases where a complete classification of trilinear forms on a vector space $V$ is available, namely $\dim(V)\leq 6$ with $\mathbb{K}$ arbitrary (see [Revoy79]) and $\dim(V)=7$ with $\mathbb{K}$ a perfect field with cohomological dimension at most $1$ (see [CH88]).

We shall briefly recall the definition of geometry of poles in the next section and we will state our main results in Section 1.2. The organization of the paper is outlined in Section 1.3.

1.1 The geometry of poles

Assume $k=3$ and let $H$ be a given hyperplane of $\mathcal{G}_{3}(V)$. For any (possibly empty) projective space $[X]$ we shall denote by $\dim(X)$ the vector dimension of $X.$

Let $[p]$ be a point of $\mathrm{PG}(V)$ and consider the point-line geometry ${\mathcal{S}}_{p}(H)$ having as points, the set of lines of $\mathrm{PG}(V)$ through $[p]$ and as lines, the set of planes $[\pi]$ of $\mathrm{PG}(V)$ through $[p]$ with $\pi\in H$. It is easy to see (see [ILP17]) that ${\mathcal{S}}_{p}(H)$ is a polar space of symplectic type (possibly a trivial one). Let $R_{p}(H):=\mathrm{Rad}\,({\mathcal{S}}_{p}(H))$ be the radical of ${\mathcal{S}}_{p}(H)$ and put $\delta(p):=\dim(\mathrm{Rad}\,({\mathcal{S}}_{p}(H))).$

We call $\delta(p)$ the degree of $[p]$ (relative to $H$). If $\delta(p)=0$ then we say that $[p]$ is smooth, otherwise we call $[p]$ a pole of $H$ or, also, a $H$-pole for short. Clearly, a point is a pole if and only if it belongs to a line of the upper radical $R^{\uparrow}(H)$ of $H$. So, $R^{\uparrow}(H)=\emptyset$ if and only if all points are smooth.

As the vector space underlying the symplectic polar space ${\mathcal{S}}_{p}(H)$ has dimension $n-1$, $\delta(p)$ is even when $n$ is odd and it is odd if $n$ is even. In particular, when $n$ is even all the points are poles of degree at least $1$. If they all have degree $1$, then we say that $H$ is spread-like.

We shall provide in Theorem 2 a direct geometric proof of the result of [ILP17, Theorem 4], stating that for $n=6$ there are spread-like hyperplanes if and only if the field $\mathbb{K}$ is not quadratically closed, extending a result of [DS14]; this is also implicit in the classification of [Revoy79]; observe that for $n=8$ we prove in [ILP17] that for quasi-algebraically closed fields there are no spread-like hyperplanes.

More in general, when all points of $\mathrm{PG}(V)$ are poles of the same degree $\delta$ and $(\delta+1)|n$, the set $\{\pi_{p}:[p]\in\mathrm{PG}(V)\}$ of all subspaces $\pi_{p}:=\{[u]\in[\ell]:p\in\ell\mbox{ and }\ell\in R^{\uparrow}(H)\}$ as $[p]$ varies in $\mathrm{PG}(V)$ might in some case possibly be a spread of $\mathrm{PG}(V)$ in spaces of projective dimension $\delta$. We are not currently aware of any case where this happens for $\delta>1$ and we propose this as a problem which might be worthy of further investigation.

1.2 Main results

Before stating our main results we need to fix a terminology for linear functionals on $\bigwedge^{3}V$ and to recall what is currently known from the literature regarding their classification. As already pointed out, a classification of alternating trilinear forms of $V$ would determine a classification of the geometries of poles defined by hyperplanes of $\mathcal{G}_{3}(V).$

We will first introduce the notion of isomorphism for hyperplanes and the notions of equivalence, near equivalence and geometrical equivalence for $k$-linear forms in general.

We say that two hyperplanes $H$ and $H^{\prime}$ of $\mathcal{G}_{k}(V)$ are isomorphic, and we write $H\cong H^{\prime}$, when $H^{\prime}=g(H):=\{g(X)\}_{X\in H}$ for some $g\in\mathrm{GL}(V)$, where $g(X)$ is the natural action of $g$ on the subspace $X$, i.e. $g(X)=\langle g(v_{1}),\dots,g(v_{k})\rangle$ for $X=\langle v_{1},\dots,v_{k}\rangle.$

Recall that two alternating $k$-linear forms $\chi$ and $\chi^{\prime}$ on $V$ are said to be (linearly) equivalent when

[TABLE]

for some $g\in\mathrm{GL}(V)$. Accordingly, if $H$ and $H^{\prime}$ are the hyperplanes associated to $\chi$ and $\chi^{\prime}$, we have $H\cong H^{\prime}$ if and only if $\chi^{\prime}$ is proportional to a form equivalent to $\chi$. Note that if $\chi^{\prime}=\lambda\cdot\chi$ for a scalar $\lambda\neq 0$ then $\chi$ and $\chi^{\prime}$ are equivalent if and only if $\lambda$ is a $k$-th power in $\mathbb{K}$.

We say that two forms $\chi$ and $\chi^{\prime}$ are nearly equivalent, and we write $\chi\sim\chi^{\prime}$, when each of them is equivalent to a non-zero scalar multiple of the other. Hence $H\cong H^{\prime}$ if and only if $\chi\sim\chi^{\prime}$.

We extend the above terminology to linear functionals of $\bigwedge^{k}V$ in a natural way, saying that two linear functionals $h,h^{\prime}\in(\bigwedge^{k}V)^{*}$ are nearly equivalent and writing $h\sim h^{\prime}$ when their corresponding $k$-alternating forms are nearly equivalent.

We say that two hyperplanes $H$ and $H^{\prime}$ are geometrically equivalent if the incidence graphs of their geometries of poles are isomorphic; the forms defining geometrically equivalent hyperplanes are called geometrically equivalent as well.

Note that nearly equivalent forms are always geometrically equivalent but the converse does not hold in general. For example, let $V$ be a vector space over a field $\mathbb{K}$ which is not quadratically closed and suppose $\dim(V)=6$. To any quadratic extension of $\mathbb{K}$ there correspond a Desarguesian line-spread $\mathcal{S}$ of $\mathrm{PG}(V)$, and the geometry $(\mathrm{PG}(V),\mathcal{S})$ is a geometry of poles associated to a trilinear form. All hyperplanes inducing line-spreads are geometrically equivalent. If $\mathbb{K}$ is a finite field or $\mathbb{K}={\mathbb{R}}$, it is easy to see that the hyperplanes inducing $\mathcal{S}$ must also be isomorphic. However, this is not the case when $\mathbb{K}={\mathbb{Q}}$ or when $\mathbb{K}$ is a field of characteristic $2$ which is not perfect. In particular, in the latter case hyperplanes arising from forms of type $T_{10,\lambda}^{(1)}$ and $T_{10,\lambda}^{(2)}$, see Table 1, are geometrically equivalent but not isomorphic.

Clearly, nearly equivalent or isomorphic hyperplanes are always geometrically equivalent.

1.2.1 Types for linear functionals of $\bigwedge^{3}V$

Given a non-trivial linear functional $h\in(\bigwedge^{3}V)^{*}$, let $\chi_{h}$ and $H_{h}$ be respectively the alternating trilinear form and the hyperplane of ${\mathcal{G}}_{3}(V)$ associated to it. When no ambiguity might arise, we shall feel free to drop the subscript $h$ in our notation.

By definition, $R_{\downarrow}(H)=[\mathrm{Rad}\,(\chi)]$, where $\mathrm{Rad}\,(\chi)=\{v\in V\colon\chi(x,y,v)=0,\forall\,x,y\in V\}.$ Define the rank of $h$ as $\mathrm{rank}\,(h):=\operatorname{cod}_{V}(\mathrm{Rad}\,(\chi))=\dim(V/\mathrm{Rad}\,(\chi))$. Obviously, functionals of different rank can never be nearly equivalent.

It is known that if $h$ is a non-trivial trilinear form, then $\mathrm{rank}\,(h)\geq 3$ and $\mathrm{rank}\,(h)\neq 4$ (see [ILP17, Proposition 19] for the latter result).

Fix now a basis $E:=(e_{i})_{i=1}^{n}$ of $V$. The dual basis of $E$ in $V^{*}$ is $E^{*}:=(e^{i})_{i=1}^{n}$, where $e^{i}\in V^{*}$ is the linear functional such that $e^{i}(e_{j})=\delta_{i,j}$ (Kronecker symbol). The set $(e^{i}\wedge e^{j}\wedge e^{k})_{1\leq i<j<k\leq n}$ is the basis of $(\bigwedge^{3}V)^{*}$ dual of the basis $(e_{i}\wedge e_{j}\wedge e_{k})_{1\leq i<j<k\leq n}$ of $\bigwedge^{3}V$ canonically associated to $E$. We shall adopt the convention of writing $\underline{ijk}$ for $e^{i}\wedge e^{j}\wedge e^{k}$, thus representing linear functionals of $\bigwedge^{3}V$ as linear combinations of symbols like $\underline{ijk}$.

In Table 1, see Appendix A, we list a number of possible types of linear functionals of $\bigwedge^{3}V$ of rank at most $7$, denoted by the symbols $T_{1},\ldots,T_{9}$ and $T^{(1)}_{10,\lambda}$, $T^{(2)}_{10,\lambda}$, $T^{(1)}_{11,\lambda}$ and $T^{(2)}_{11,\lambda}$, $T_{12,\mu}$ where $\lambda$ is a scalar subject to the conditions specified in the table. Whenever $T$ is one of the types of Table 1, we say that $h\in(\bigwedge^{3}V)^{*}$ is of type $T$ if $h$ is nearly equivalent to the linear functional described at row $T$ of Table 1. The type of $H_{h}$ or $\chi_{h}$ is the type of $h$. By definition, functionals of the same type are nearly equivalent. Note that the definitions of each of these types make sense for any $n$ and for any field $\mathbb{K}$, provided that $n$ is not smaller than the rank of (a linear functional admitting) that description and that $\mathbb{K}$ contains elements satisfying the special conditions there outlined.

In particular, Table 1 provides a complete classification (up to equivalence) in the case of perfect fields of cohomological dimension at most $1$ and $\dim(V)\leq 7$, see [CH88, Revoy79]. We recall that in [CH88] the authors also determine the full automorphism group associated to each form.

It is well known that a general classification of trilinear forms up to equivalence is hopeless; for instance for $\mathbb{K}={\mathbb{C}}$ and $n=9$ there are infinite families of linearly inequivalent trilinear forms; see e.g. [DS10].

When $n\leq 6$, Revoy [Revoy79] proves that all trilinear forms, up to equivalence, are of type $T_{i}$, $1\leq i\leq 4$, $T_{10,\lambda}^{(1)}$, $T_{10,\lambda}^{(2)}$. In particular, if the field is quadratically closed, all forms of rank $6$ are either of type $T_{3}$ or of type $T_{4}$. If the field is not quadratically closed, it is possible to distinguish two families of classes of forms linearly equivalent among themselves according as they are equivalent to $T_{3}$ or to $T_{4}$ over the quadratic closure $\mathbb{K}^{\square}$ of $\mathbb{K}$. More in detail, if $\mathrm{char}\,(\mathbb{K})$ is odd, $T_{10,\lambda}^{(1)}=T_{10,\lambda}^{(2)}$ and each form of a type in $T_{10,\lambda}^{(1)}$ is equivalent to $T_{3}$ over $\mathbb{K}^{\square}$. If $\mathrm{char}\,(\mathbb{K})=2$, then the classes $T_{10,\lambda}^{(2)}$ and $T_{10,\lambda}^{(1)}$ are in correspondence with respectively the separable and the inseparable quadratic extensions of the field $\mathbb{K}$; furthermore, any form of a type in $T_{10,\lambda}^{(2)}$ is equivalent to a form of type $T_{3}$ in $\mathbb{K}^{\square}$, while any form of type $T_{10,\lambda}^{(1)}$ is equivalent in $\mathbb{K}^{\square}$ to a form of type $T_{4}$.

If $n=7$ and $\mathbb{K}$ is a perfect field of cohomological dimension at most $1$, Cohen and Helmick [CH88] show that all trilinear forms, up to equivalence, are of a type described in Table 1.

Under the conditions of Table 1, we shall provide a classification for the geometries of poles.

We now present our main results; for the notions of extension, expansion and block decomposition as well as some of the notation, see Section 3. By the symbol $|x,y|_{ij}$ we mean the $(i,j)$-Plücker coordinate of the line $[x,y]$ spanned by the vectors $x=(x_{i})_{i=1}^{n}$ and $y=(y_{i})_{i=1}^{n}$ written in coordinates with respect to the basis $E$, i.e. $|x,y|_{ij}:=x_{i}y_{j}-x_{j}y_{i}$ is the $ij$-coordinate of $e_{i}\wedge e_{j}$ with respect to the basis $(e_{i}\wedge e_{j})_{1\leq i<j\leq n}$.

Theorem 1**.**

Suppose $\dim(V)\leq 6$ and let $h$ be a non-trivial linear functional on $\bigwedge^{3}V$ having type as described in Table 1, with associated alternating trilinear form $\chi$. Denote by $H$ the hyperplane of $\mathcal{G}_{3}(V)$ defined by $h$. Then one of the following occurs:

$h$* has type $T_{1}$ (rank $3$). In this case $H$ is the trivial hyperplane centered at $\mathrm{Rad}\,(\chi)$ and $R^{\uparrow}(H)$ is the set of the lines of $\mathrm{PG}(V)$ that meet $[\mathrm{Rad}\,(\chi)]$ non-trivially.* 2. 2.

$h$* has type $T_{2}$ (rank $5$), namely $\mathrm{Rad}\,(\chi)$ is $1$-dimensional. In this case $H$ is a trivial extension $\mathrm{Ext}_{\mathrm{Rad}\,(\chi)}(\mathrm{Exp}(H_{0}))$ of a symplectic hyperplane $\mathrm{Exp}(H_{0})$, constructed in a complement $V_{0}$ of $\mathrm{Rad}\,(\chi)$ in $V$ starting from the line-set $H_{0}$ of a symplectic generalized quadrangle. The elements of $R^{\uparrow}(H)$ are the lines of $\mathrm{PG}(V)$ that either belong to $H_{0}$ or pass through the point $[\mathrm{Rad}\,(\chi)]=R_{\downarrow}(H)$ or such that their projection onto $V_{0}$ is in $H_{0}.$* 3. 3.

$h$* has type $T_{3}$ (rank $6$). Then $H$ is a decomposable hyperplane $\mathrm{Dec}(H_{0},H_{1})$ arising form the hyperplanes $H_{0}$ and $H_{1}$ of $\mathcal{G}_{3}(V_{0})$ and $\mathcal{G}_{3}(V_{1})$ for a suitable decomposition $V=V_{0}\oplus V_{1}$ with $\dim(V_{0})=\dim(V_{1})=3$. Then $R^{\uparrow}(H)=\{[x,y]\colon x\in V_{0}\setminus\{0\},~{}y\in V_{1}\setminus\{0\}\}.$* 4. 4.

$h$* has type $T_{4}$ (rank $6$). Then $R^{\uparrow}(H)=\{[x,y]=[a+b,\omega(a)]\colon a\in V_{0}\setminus\{0\},b\in V_{1}\}\cup\{[x,y]\subseteq V_{1}\}$ for a decomposition $V=V_{0}\oplus V_{1}$ with $\dim(V_{0})=\dim(V_{1})=3$ and $\omega$ an isomorphism of $V$ interchanging $V_{0}$ and $V_{1}$.* 5. 5.

$h$* has type $T^{(1)}_{10,\lambda}$ or $T^{(2)}_{10,\lambda}$ (rank $6$). Then $R^{\uparrow}(H)$ is a Desarguesian line spread of $\mathrm{PG}(V)$ corresponding to the field extension $\mathbb{K}[\mu]$ with $\mu$ a root of $p_{\lambda}(t)$.*

Theorem 2**.**

Let $V:=V(6,\mathbb{K}).$ Line-spreads of $\mathrm{PG}(V)$ induced by hyperplanes of $\mathcal{G}_{3}(V)$ exist if and only if $\mathbb{K}$ is a non-quadratically closed field.

Draisma and Shaw prove that when $\mathbb{K}$ is a finite field there always exist hyperplanes of $\mathcal{G}_{3}(V)$ with $\dim V=6$ having a Desarguesian spread as upper radical [DS14, §3.1 and §3.2], while these hyperplanes do not exist when $\mathbb{K}$ is algebraically closed with characteristic [math], [DS14, Remark 9].

Our Theorem 2 generalizes their results to arbitrary fields $\mathbb{K}$ and provides necessary and sufficient conditions for the existence of spread-like hyperplanes for $n=6$; its statement correspond to Theorem 20, point 5 in [ILP17] (and also to Theorem 4 in [ILP17]).

It also further clarifies the result of [Revoy79] linking the forms $T_{10,\lambda}^{(i)}$ with quadratic extensions of the field $\mathbb{K}$.

Theorem 3**.**

Suppose $\dim(V)=7$ and let $h$ be a non-trivial linear functional on $\bigwedge^{3}V$ having type as described in Table 1. Denote by $\chi$ the associated alternating trilinear form and by $H$ the hyperplane of $\mathcal{G}_{3}(V)$ defined by $h$.

If $\mathrm{rank}\,(h)\leq 6$ then $H$ is a trivial extension $\mathrm{Ext}_{\mathrm{Rad}\,(h)}(H^{\prime})$ where $H^{\prime}$ is a hyperplane of $\mathcal{G}_{3}(V^{\prime})$ with $V=\mathrm{Rad}\,(h)\oplus V^{\prime}$, $\dim(V^{\prime})\leq 6$, and $H^{\prime}$ is defined by a trilinear form $h^{\prime}$ of type as in Theorem 1.

If $\mathrm{rank}\,(h)=7$ then one of the following occurs:

$h$* has type $T_{5}$. Then, there are two non-degenerate symplectic polar spaces ${\mathcal{S}}_{1}$ and ${\mathcal{S}}_{2}$, embedded as distinct hyperplanes in $\mathrm{PG}(V)$ such that both determine the same polar space ${\mathcal{S}}_{0}$ on their intersection. The radical of ${\mathcal{S}}_{0}$ is a point, say $p_{0}$. There are also two totally isotropic planes $A_{1}$ and $A_{2}$ of ${\mathcal{S}}_{0}$ such that $A_{1}\cap A_{2}=\{p_{0}\}$. The poles of $H$ are the points of $\mathcal{S}_{1}\cup\mathcal{S}_{2}$, the poles of degree $4$ being the points of $A_{1}\cup A_{2}$. The lines of ${\mathcal{P}}(H)$ are the totally isotropic lines of ${\mathcal{S}}_{i}$ that meet $A_{i}$ non-trivially, for $i=1,2$.* 2. 2)

$h$* has type $T_{6}$. The poles of $H$ lie in a hyperplane $\mathcal{S}$ of $\mathrm{PG}(V)$. A non-degenerate polar space of symplectic type is defined in $\mathcal{S}$ and a totally isotropic plane $A$ of $\mathcal{S}$ is given. The lines of ${\mathcal{P}}(H)$ are the totally isotropic lines of $\mathcal{S}$ that meet $A$ non-trivially. The points of $A$ are the poles of $H$ of degree $4$.* 3. 3)

$h$* has type $T_{7}$. The poles of $H$ are the points of a cone of $\mathrm{PG}(V)$ having as vertex a plane $A$ and as basis a hyperbolic quadric $\mathcal{Q}$. A conic $\mathcal{C}$ is given in $A$ such that the elements of $\mathcal{C}$ are the poles of degree $4$. There is a correspondence mapping each point $[x]\in\mathcal{C}$ to a line $\ell_{x}$ contained in a regulus of $\mathcal{Q}$. For each $[x]\in\mathcal{C}$ it is possible to define a line spread $\mathcal{S}_{x}$ of $\langle A,\ell_{x}\rangle/\langle x\rangle\cong\mathrm{PG}(3,\mathbb{K})$ such that $R^{\uparrow}(H)=\{\ell\subseteq\langle x,s\rangle:[x]\in\mathcal{C},s\in\mathcal{S}_{x}\}$.* 4. 4)

$h$* has type $T_{8}$. In this case $H=\mathrm{Exp}(H_{0})$ is a symplectic hyperplane. In particular, the geometry ${\mathcal{P}}(H)$ is a non-degenerate polar space of symplectic type and rank $3$, naturally embedded in a hyperplane $[V_{0}]$ of $\mathrm{PG}(V)$. All poles of $H$ have degree $4$.* 5. 5)

$h$* has type $T_{9}$. Then ${\mathcal{P}}(H)$ is a split-Cayley hexagon naturally embedded in a non-singular quadric of $\mathrm{PG}(V)$. All poles of $H$ have degree $2$.* 6. 6)

$h$* has type $T^{(i)}_{11,\lambda}$, $i=1,2$. The poles of $H$ are the points of a subspace $\mathcal{S}$ of codimension $2$ in $V$. There is only one point $[p]\in\mathcal{S}$ which is a pole of degree $4$. Furthermore, there is a line-spread $\mathcal{F}$ of $\mathcal{S}/\langle p\rangle\cong\mathrm{PG}(3,\mathbb{K})$ such that $R^{\uparrow}(H):=\{\ell\subseteq[\pi,p]:\pi\in\mathcal{F}\}.$*

Theorems 1 and 3 correspond to Theorems 20 and 21 of [ILP17], where they were presented without a detailed proof. In the present paper we have chosen to refine the results announced [ILP17], by providing a fully geometric description of the geometries of poles arising for $n\leq 7$, without having to recourse to coordinates. In any case, the original statements for cases 3,5 and 7 of [ILP17, Theorem 21] can be immediately deduced from Theorem 3 in light of the equations of Table 5.

1.3 Organization of the paper

In Section 2 we shall explain how to algebraically describe points and lines of a geometry of poles. Draisma and Shaw [DS10] have shown that either the set of $H$-poles is all of $\mathrm{PG}(V)$ or it determines an algebraic hypersurface in $\mathrm{PG}(V)$ described by an equation of degree $(n-3)/2$. In Section 2 we shall study such varieties. In particular, in Section 2.2 we shall explicitly determine their equations as determinantal varieties and in Section 2.3 describe some hyperplanes whose variety of poles is reducible in the product of distinct linear factors. In Section 3 we will present three families of hyperplanes of $\mathcal{G}_{k}(V)$ obtained by extension, expansion and block decomposable construction. Our main theorems will be proved in Section 4. For the ease of the reader, all the tables are collected in Appendix A.

2 Geometry of poles

Throughout this section we take $E=(e_{i})_{1\leq i\leq n}$ as a given basis of $V$ and the coordinates of vectors of $V$ will be given with respect to $E.$

2.1 Determination of points and lines

Let $h\colon\bigwedge^{3}V\rightarrow\mathbb{K}$ be a linear functional associated to a given hyperplane $H$ of $\mathcal{G}_{3}(V)$, where $V$ is a $n$-dimensional vector space over a field $\mathbb{K}.$

For any $u\in V$ consider the bilinear alternating form

[TABLE]

By definition of $H$-pole, a point $[u]\in\mathrm{PG}(V)$ is a $H$-pole if and only if the radical of $h_{u}$ is not trivial. Consider also the bilinear alternating form on $V$

[TABLE]

where $M_{u}$ is the matrix associated to $\chi_{u}$ with respect to the basis $E$ of $V$. Clearly, $\mathrm{Rad}\,(h_{u})=(\mathrm{Rad}\,(\chi_{u}))/\langle u\rangle$; thus the rank of the matrix of $h_{u}$ with respect to any basis of $V/\langle u\rangle$ and the rank of the matrix $M_{u}$ are exactly the same.

Proposition 2.1**.**

Let $[u]$ be a point of $\mathrm{PG}(V)$ with $u=(u_{i})_{i=1}^{n}$ and let $M_{u}$ be the $n\times n$-matrix associated to the alternating bilinear form $\chi_{u}$. If $u_{i}\not=0$ then the $i$-th column (row) of $M_{u}$ is a linear combination of the other columns (rows) of $M_{u}.$

Proof.

Denote by $C_{1},\dots C_{n}$ the columns of the matrix $M_{u}$ and let $x=(x_{i})_{i=1}^{n}$ and $u=(u_{i})_{i=1}^{n}.$ Let $M_{u}^{(i)}$ be the $(n-1)\times(n-1)$-submatrix of $M_{u}$ obtained by deleting its $i$-th column and $i$-th row. For any $x\in V,$ the condition $x^{T}M_{u}u=0$ is equivalent to

[TABLE]

where $x^{T}\cdot C_{i}:=\sum_{j=1}^{n}x_{j}c_{ji}$ with $C_{i}=(c_{ji})_{j=1}^{n}.$ Since $u_{i}\not=0$, we have

[TABLE]

i.e.

[TABLE]

The previous condition holds for any $x\in V,$ hence

[TABLE]

As $M_{u}$ is antisymmetric, the same argument can be applied also to the $i$-th row of $M_{u}.$ ∎

The following corollaries are straightforward.

Corollary 2.2**.**

$\mathrm{rank}\,(M_{u})\leq n-1.$**

Corollary 2.3**.**

The point $[u]$ is a $H$-pole if and only if $\mathrm{rank}\,(M_{u})\leq n-2.$

Note that the matrix $M_{u}$ is antisymmetric; hence its rank must be an even number. By Corollary 2.3 it is clear that if $n$ is even then every point of $\mathrm{PG}(V)$ is a $H$-pole and this holds for any hyperplane $H$ of $\mathcal{G}_{3}(V).$ More precisely, the degree of the point $[u]$ is $\delta(u)=(n-1)-\mathrm{rank}\,(M_{u}).$ So, it is straightforward to see that the set of all the $H$-poles of degree at least $t$ is either the whole of $\mathrm{PG}(V)$ or the determinantal variety of $\mathrm{PG}(V)$ described by the condition $\mathrm{rank}\,(M_{u})\leq(n-1)-t\leq n-2$; see [harris, Lecture 9] for some properties of these varieties. Furthermore all entries of $M_{u}$ are linear homogeneous polynomials in the coordinates of $u$; so the condition $\mathrm{rank}\,(M_{u})\leq n-2$ provides algebraic conditions on the coordinates of $u$ for $[u]$ to be a $H$-pole.

In Table 2 and Table 3 of Appendix A we have explicitly written the matrices $M_{u}$ associated to the trilinear forms $h$ of type $T_{i}$ of Table 1 where $u=(u_{i})_{i=1}^{n}.$ To simplify the notation, when $\mathrm{rank}\,(h)<\dim(V)$, we have just written the $\mathrm{rank}\,(h)\times\mathrm{rank}\,(h)$-matrix associated to $h_{u}|_{V/\mathrm{Rad}\,(h)}.$

To determine the elements of the upper radical of $H$, namely the lines $\ell=[x,y]$ of $\mathrm{PG}(V)$ with the property that any plane through them is in $H$, we need to determine conditions on $x$ and $y$ such that the linear functional

[TABLE]

is null. To do this, it is sufficient to require that $\tilde{h}_{xy}$ annihilates on the basis vectors of $V$, i.e. $\tilde{h}_{xy}(e_{i})=0,$ for every $i=1,\dots,n$.

2.2 A determinantal variety

Let $h$ be a trilinear form associated to the hyperplane $H$ and for any $u\in V$, let $\chi_{u}$ be the alternating bilinear form as in Equation (1) whose representative matrix is $M_{u}.$ With $1\leq i\leq n$, denote by $M_{u}^{(i)}$ the principal submatrix of $M_{u}$ obtained by deleting its $i$-th row and its $i$-th column. The matrix $M_{u}^{(i)}$ is a $(n-1)\times(n-1)$-antisymmetric matrix whose entries are linear functionals defined over $\mathbb{K}$; so, its determinant is a polynomial of degree $n-1$ in the unknowns $u_{1},\ldots,u_{n}$ which is a square in the ring $\mathbb{K}[u_{1},\ldots,u_{n}]$, that is there exists a polynomial $d_{i}(u_{1},\ldots,u_{n})$ with $\deg d_{i}(u_{1},\ldots,u_{n})=(n-1)/2$ such that

[TABLE]

Define $g_{i}(u_{1},\ldots,u_{n})$ to be the polynomial in $\mathbb{K}[u_{1},\ldots,u_{n}]$ such that

[TABLE]

where $\alpha\in{\mathbb{N}}$ and $u_{i}^{\alpha_{i}+1}$ does not divide $d_{i}(u_{1},\ldots,u_{n})$.

Theorem 2.4**.**

The set $P(H)$ of $H$-poles is either the whole pointset of $\mathrm{PG}(V)$ or there exists an index $i$, $1\leq i\leq n$, such that $P(H)$ is an algebraic hypersurface of $\mathrm{PG}(V)$ with equation $g_{i}(u_{1},\ldots,u_{n})=0$.

Proof.

Suppose $P(H)\neq\mathrm{PG}(V)$. By [DS10], $P(H)$ is an algebraic hypersurface admitting an equation of degree $(n-3)/2$; so $P(H)$ contains at most $(n-3)/2$ coordinate hyperplanes of the form $\Pi_{j}:u_{j}=0$. Then, there exists $i$ with $1\leq i\leq n$ such that $\Pi_{i}\not\subseteq P(H)$. By Corollary 2.3, $[u]\in\mathrm{PG}(V)$ is a $H$-pole if and only if $\mathrm{rank}\,(M_{u})\leq n-2.$ If $n$ is even, this always happens. So, assume $n$ odd. We shall work over the algebraic closure $\overline{\mathbb{K}}$ of $\mathbb{K}$. Take $u=(u_{1},\ldots,u_{n})$ with $u_{j}\in\overline{\mathbb{K}}$. We can regard $u$ as a vector in coordinates with respect to the basis induced by $E$ on $\overline{V}:=V\otimes\overline{\mathbb{K}}$. In any case, the matrix $M_{u}$ is antisymmetric and its entries are homogeneous linear functionals in $u_{1},\ldots,u_{n}$ defined over the field $\mathbb{K}$.

Consider the points $[u]$ with $u_{i}\not=0.$ By Proposition 2.1, the $i$-th row/column of $M_{u}$ is a linear combination of the remaining $n-1$ rows/columns. So, $\mathrm{rank}\,M_{u}=\mathrm{rank}\,M_{u}^{(i)}$.

Let $d_{i}(u_{1},\ldots,u_{n})$ be as in Equation (3). We now show that

[TABLE]

for all $u_{1},\ldots,u_{i-1},u_{i+1},\ldots,u_{n}\in\overline{\mathbb{K}}$. Indeed, when $u_{i}=0$, by Proposition 2.1, there exists a column $C_{j}$ of $M_{u}$, $u_{j}\not=0$, such that

[TABLE]

So, the $j$-th column of $M_{u}^{(i)}$ is also a linear combination of the other columns of $M_{u}^{(i)}.$ Hence $\det M_{u}^{(i)}=0=(d_{i}(u_{1},\ldots,0,\ldots,u_{n}))^{2}$.

Since $\overline{\mathbb{K}}$ is algebraically closed, we have

[TABLE]

with $d^{\prime}_{i}(u_{1},\ldots,u_{n})$ a polynomial in $\mathbb{K}[u_{1},\ldots,u_{n}]$ with $\deg d^{\prime}_{i}=(n-3)/2$. We remark that the unknowns of the polynomials may assume their values in $\overline{\mathbb{K}}$ but the coefficients are all in $\mathbb{K}.$ By Corollary 2.3, a point $[u]\in\mathrm{PG}(\overline{V})\setminus\Pi_{i}$ (i.e. $u_{i}\neq 0$) is a $H$-pole if and only if $\det M_{u}^{(i)}=0$, i.e. $d^{\prime}_{i}(u_{1},\ldots,u_{n})=0.$

Denote by $\overline{\Gamma}_{i}$ the algebraic variety (over $\overline{\mathbb{K}}$) of equation $d_{i}^{\prime}(u_{1},\ldots,u_{n})=0$. Since we are assuming $\Pi_{i}\not\subseteq P(H)$ and $P(H)\setminus\Pi_{i}=\overline{\Gamma}_{i}\setminus\Pi_{i}$, we have

[TABLE]

where $C(X)$ denotes the projective closure of $X$ in $\mathrm{PG}(\overline{V})$ with $\Pi_{i}$ regarded as the hyperplane at infinity. Note that since $P(H)$ is an algebraic variety of $\mathrm{PG}(\overline{V})$ which does not contain $\Pi_{i}$, the points of $P(H)\cap\Pi_{i}$ are exactly those of the projective closure $C(P(H)\setminus\Pi_{i})\cap\Pi_{i}$. The same applies to the points at infinity of the affine variety $\overline{\Gamma}_{i}\setminus\Pi_{i}$.

Suppose $u_{i}^{\beta}|d_{i}^{\prime}(u_{1},\ldots,u_{n})$ and $u_{i}^{\beta+1}\not|d_{i}^{\prime}(u_{1},\ldots,u_{n})$ for $\beta\in{\mathbb{N}}$.

Then, $d_{i}^{\prime}(u_{1},\ldots,u_{n})=u_{i}^{\beta}g_{i}(u_{1},\ldots,u_{n})$ and $g_{i}(u_{1},\ldots,u_{n})=0$ is an equation for $C(\overline{\Gamma}_{i}\setminus\Pi_{i})=P(H)$. This completes the proof. ∎

Observe that if $\alpha_{i}=1$, then $g_{i}(u_{1},\ldots,u_{n})=0$ is exactly an equation of degree $(n-3)/2$ for $P(H)$.

If $n$ is even then each point of $\mathrm{PG}(V)$ is a $H$-pole. Likewise, when $n$ is odd and $H$ is defined by a trilinear form $\chi$ of rank less than $n$, then also each point of $\mathrm{PG}(V)$ is a $H$-pole. Indeed, whenever $R_{\downarrow}(H)=\mathrm{Rad}\,(\chi)$ is non-trivial, for any $[u]\in\mathrm{PG}(V)$ there exists a line $\ell$ through $[u]$ in $R^{\uparrow}(H)$ which meets $\mathrm{Rad}\,(\chi)$ and so $[u]$ is a $H$-pole. In any case the above conditions, while sufficient, are not in general necessary; in fact, in Section 3.3 we shall provide a construction which might lead to alternating forms of rank $n$ with $n$ odd and $R_{\downarrow}(H)=\emptyset$ such that all points of $\mathrm{PG}(V)$ are $H$-poles.

Note that Theorem 2.4 shows that the variety of the $H$-poles in $\mathrm{PG}(V)$ admits at least one equation over $\mathbb{K}$ of degree at most $(n-3)/2$; this does not mean that $(n-3)/2$ is the minimum degree for such an equation or that the polynomial $g_{i}(u_{1},\ldots,u_{n})$ generates the radical ideal of such variety (even over $\mathbb{K}$). For instance, in the case of symplectic hyperplanes, see Section 3.2.1, the variety of poles is always a hyperplane of $\mathrm{PG}(V)$ and so it admits an equation of degree $1$ for any odd $n$.

2.3 A family of reducible hyperplanes

It is an interesting problem to investigate which algebraic varieties might arise as set of $H$-poles; as noted before such varieties will always be skew-symmetric determinantal varieties [Ha]. We leave the development of this study to a further paper. However, with this aim in mind, we give here a construction for hyperplanes whose variety of poles are reducible.

Theorem 2.5**.**

Suppose $V=V_{1}+V_{2}$ with $(e_{1},\ldots,e_{n_{1}})$ and $(e_{n_{1}},\ldots,e_{n})$ bases of $V_{1}$ and $V_{2}$ respectively. For $i=1,2$ let $H_{i}$ be a hyperplane of $\mathcal{G}_{3}(V_{i})$ whose $H_{i}$-poles in $\mathrm{PG}(V_{i})$ satisfy respectively the equations $f_{1}(u_{1},\ldots,u_{n_{1}})=0$ and $f_{2}(u_{n_{1}},\ldots,u_{n})=0$. Then there exists a hyperplane $H$ of $\mathcal{G}_{3}(V)$ whose set of $H$-poles defines a variety of $\mathrm{PG}(V)$ with equation $f(u_{1},\ldots,u_{n})=0$ where

[TABLE]

Proof.

For $i=1,2$ denote by $\overline{h}_{i}$ the trilinear form on $V_{i}$ defining $H_{i}$ and consider the extension $h_{i}\colon V\times V\times V\rightarrow\mathbb{K}$ given by

[TABLE]

where $\pi_{1}\colon V\rightarrow V_{1}$ is the projection on $V_{1}$ along $\langle e_{n_{1}+1},\dots e_{n}\rangle$ and $\pi_{2}\colon V\rightarrow V_{2}$ is the projection on $V_{2}$ along $\langle e_{1},\dots e_{n_{1}-1}\rangle.$

Put $h:=h_{1}+h_{2}$ and let $H$ be the hyperplane of $\mathcal{G}_{3}(V)$ defined by $h.$ For any $[u]\in\mathrm{PG}(V)$, let $\chi_{u}$ as in Equation (1) be the bilinear alternating form induced by $h$ and represented by the matrix $M_{u}$ with respect to the basis $(e_{1},\ldots,e_{n_{1}},\ldots,e_{n})$ of $V.$ By construction, the matrix $M_{u}$ has the following structure:

[TABLE]

where $\bar{M}_{1}:=\begin{pmatrix}M_{1}&(v^{1}_{n_{1}})\\ -{(v^{1}_{n_{1}})}^{T}&0\end{pmatrix}$ is the $n_{1}\times n_{1}$-matrix representing $\overline{h}_{1}$ and $\bar{M}_{2}=\begin{pmatrix}0&-{(v^{2}_{n_{1}})}^{T}\\ (v^{2}_{n_{1}})&M_{2}\\ \end{pmatrix}$ is the $(n-n_{1}+1)\times(n-n_{1}+1)$-matrix representing $\overline{h}_{2}.$ Note that $(v^{1}_{n_{1}})$ and $(v^{2}_{n_{1}})$ are suitable columns with entries respectively in the rings $\mathbb{K}[u_{1},\ldots,u_{n_{1}}]$ and $\mathbb{K}[u_{n_{1}},\ldots,u_{n}]$.

Suppose $u_{n_{1}}=0$. Since the entries of $\bar{M}_{1}$ and $\bar{M}_{2}$ are respectively linear functionals in $u_{1},\ldots,u_{n_{1}}$ and $u_{n_{1}},\ldots,u_{n}$ if all entries $u_{i}$ of $u$ with $i\geq n_{1}$ are null, then $\bar{M}_{2}$ is a zero matrix and $\mathrm{rank}\,M_{u}\leq n-2$. Likewise if all entries $u_{i}$ of $u$ with $i\leq n_{1}$ are zero, then $\bar{M}_{1}$ is the zero matrix and $\mathrm{rank}\,M_{u}\leq n-2$. Assume now that $u_{n_{1}}=0$ and that there exist $i,j$ with $i<n_{1}$ and $j>n_{1}$ such that $u_{i}\neq 0\neq u_{j}$. Then, by Proposition 2.1, the first $(n_{1}-1)$ columns of $\bar{M}_{1}$ and the last $n-n_{1}$ columns of $\bar{M}_{2}$ are a linearly dependent set; in particular, $\mathrm{rank}\,M_{u}\leq n-2$. So we have proved that the hyperplane $\Pi_{n_{1}}:u_{n_{1}}=0$ is contained in the variety of poles $P(H)$.

By construction, $\mathrm{rank}\,\bar{M}_{1}\leq n_{1}-2$ if and only if $f_{1}(u_{1},\ldots,u_{n_{1}})=0$ and $\mathrm{rank}\,\bar{M}_{2}\leq n-n_{1}-1$ if and only if $f_{2}(u_{n_{1}},\ldots,u_{n})=0$. Let $\Delta$ be the variety of equation $f_{1}(u_{1},\ldots,u_{n_{1}})f_{2}(u_{n_{1}},\ldots,u_{n})=0$. Observe that $\Delta\setminus\Pi_{n_{1}}\subseteq P(H)\setminus\Pi_{n_{1}}$, as $[u]\in\Delta\setminus\Pi_{n_{1}}$ implies that either $\mathrm{rank}\,\bar{M}_{1}^{(n_{1})}\leq\mathrm{rank}\,\bar{M}_{1}\leq n_{1}-2$ or $\mathrm{rank}\,\bar{M}_{2}^{(n_{1})}\leq\mathrm{rank}\,\bar{M}_{2}\leq n-n_{1}-1$ and, by Proposition 2.1 ($u_{n_{1}}\not=0$), the $n_{1}$-th column of $M$ is a linear combinations of the columns of $\bar{M}_{1}^{(n_{1})}$ as well as a linear combination of the columns of $\bar{M}_{2}^{(n_{1})}$ (so it does not contribute to the rank); so $\mathrm{rank}\,M_{u}\leq n-2$ and $[u]\in P(H)\setminus\Pi_{n_{1}}$. Conversely, suppose $[u]\in P(H)$ with $u_{n_{1}}\neq 0$. Then, by Theorem 2.4,

[TABLE]

If $[u]\in\mathrm{PG}(V)\setminus\Pi_{n_{1}}$, again by Theorem 2.4 applied to $V_{1}$ and $V_{2}$ we have $\det\bar{M}_{1}^{(n_{1})}=0$ if and only if $f_{1}(u_{1},\ldots,u_{n_{1}})=0$ and $\det\bar{M}_{2}^{(n_{1})}=0$ if and only if $f_{2}(u_{n_{1}},\ldots,u_{n})=0$. So $P(H)\setminus\Pi_{{n_{1}}}\subseteq\Delta\setminus\Pi_{n_{1}}$, whence

[TABLE]

Since $\Pi_{n_{1}}\subseteq P(H)$, we have

[TABLE]

and, consequently $f(u_{1},\ldots,u_{n})=u_{n_{1}}\cdot f_{1}(u_{1},\ldots,u_{n_{1}})\cdot f_{2}(u_{n_{1}},\ldots,u_{n})$ is an equation for $P(H)$. ∎

Corollary 2.6**.**

Let $V$ be a vector space of odd dimension $n\geq 5$ over a field $\mathbb{K}$. Then there exists a hyperplane $H$ of $\mathcal{G}_{3}(V)$ whose set of $H$-poles is the union of $(n-3)/2$ distinct hyperplanes of $\mathrm{PG}(V)$.

Proof.

We proceed by induction on $n$ odd. If $n=5$, consider the hyperplane of $\mathcal{G}_{3}(V)$ defined by the trilinear form $h:=\underline{123}+\underline{345}.$ It is easy to verify that its poles are all the points of the hyperplane of equation $u_{3}=0.$

By induction hypothesis, suppose that the thesis holds for vector spaces of odd dimension $n$. We shall prove it also holds for vector spaces of (odd) dimension $n+2.$ Let $V$ with $\dim(V)=n+2$ and let $(e_{i})_{i=1,\dots,n+2}$ be a given basis of $V.$ Put $V_{1}:=\langle e_{i}\rangle_{1\leq i\leq n}$ and $V_{2}:=\langle e_{n},e_{n+1},e_{n+2}\rangle.$ Clearly $V=V_{1}+V_{2}.$ As $\dim(V_{1})=n$ we can apply the induction hypothesis. So, there exists a trilinear form $\overline{h}_{1}$ on $V_{1}$ (defining a hyperplane $H_{1}$ of $\mathcal{G}_{3}(V_{1})$) such that its set of poles is the union of $(n-3)/2$ distinct hyperplanes of $\mathrm{PG}(V)$. Equivalently, we can assume without loss of generality that any $H_{1}$-pole satisfies the equation $g_{1}(u_{1},\ldots,u_{n})=0$ where $g_{1}(u_{1},\ldots,u_{n}):=\prod_{i=1}^{(n-3)/2}u_{2i+1}.$

Let $\overline{h}_{2}$ be the trilinear form on $V_{2}$ defined by $\overline{h}_{2}=\underline{(n)(n+1)(n+2)}$. Clearly $\overline{h}_{2}$ has no pole in $\mathrm{PG}(V_{2}).$

By (the proof of) Theorem 2.5, we can consider the hyperplane $H$ of $\mathcal{G}_{3}(V)$ defined by the sum of the extensions $h_{1}$ and $h_{2}$ to $V$ of $\overline{h}_{1}$ and $\overline{h}_{2}$ (see the beginning of the proof of Theorem 2.5 for the definition of $\overline{h}_{1}$ and $\overline{h}_{2}$). Then, the set of $H$-poles is a variety of $\mathrm{PG}(V)$ with equation $g(u_{1},\ldots,u_{n+2})=0$ where

[TABLE]

∎

3 Constructions of families of hyperplanes

In this section we will explain some general constructions yielding large families of hyperplanes of $k$-Grassmannians. More precisely, in Sections 3.1 and 3.2 we shall briefly recall (without proofs) two constructions already introduced in [ILP17] while in Section 3.3 we will present a new one.

We first need to give the following definition which extends the definition of $\mathcal{S}_{p}(H)$ given in Section 1.1. For a $(k-2)$-subspace $X$ of $V$, let $(X)G_{k}$ be the set of $k$-subspaces of $V$ containing $X$. This is a subspace of $\mathcal{G}_{k}(V)$. Let $(X)\mathcal{G}_{k}$ be the geometry induced by $\mathcal{G}_{k}(V)$ on $(X)G_{k}$ and put $(X)H:=(X)G_{k}\cap H$. Then $(X)\mathcal{G}_{k}\cong\mathcal{G}_{2}(V/X)$ and either $(X)H=(X)G_{k}$ or $(X)H$ is a hyperplane of $(X)\mathcal{G}_{k}$. In either case, the point-line geometry ${\mathcal{S}}_{X}(H)=((X)G_{k-1},(X)H)$ is a polar space of symplectic type (possibly a trivial one, when $(X)H=(X)G_{k})$). Let $R_{X}(H):=\mathrm{Rad}\,({\mathcal{S}}_{X}(H))$ be the radical of ${\mathcal{S}}_{X}(H)$.

3.1 Extensions and trivial extensions

Let $V=V_{0}\oplus V_{1}$ be a decomposition of $V$ as the direct sum of two non-trivial subspaces $V_{0}$ and $V_{1}$. Put $n_{0}:=\dim(V_{0})$ and assume that $n_{0}\geq k$ ($\geq 3$). Let $\chi_{0}:V_{0}\times\cdots\times V_{0}\rightarrow\mathbb{K}$ be a non-trivial $k$-linear alternating form on $V_{0}$. The form $\chi_{0}$ can naturally be extended to a $k$-linear alternating form $\chi$ of $V$ by setting

[TABLE]

and then extending by (multi)linearity. Let $H_{\chi}$ be the hyperplane of $\mathcal{G}_{k}(V)$ defined by $\chi$. Then, the following properties hold:

Theorem 3.1** ([ILP17]).**

Let $\chi_{0}$ be a $k$-alternating linear form on $V_{0}$ and $n_{0}=\dim V_{0}$; define $\chi$ as in (4). For $n_{0}=k$, put $H_{0}=\emptyset$; otherwise, let $H_{0}$ be the hyperplane of $\mathcal{G}_{k}(V_{0})$ defined by $\chi_{0}$. Let also $\pi:V\rightarrow V_{0}$ be the projection of $V$ onto $V_{0}$ along $V_{1}$. Then,

1. (1)

$H_{\chi}~{}=~{}\{X\in\mathcal{G}_{k}(V)~{}\colon~{}\mbox{either}~{}X\cap V_{1}\neq 0~{}\mbox{or}~{}\pi(X)\in H_{0}\}$. 2. (2)

$R_{\downarrow}(H_{\chi})~{}=~{}\langle R_{\downarrow}(H_{0})\cup[V_{1}]\rangle$*  where the span in taken in $\mathrm{PG}(V)$.* 3. (3)

$R^{\uparrow}(H_{\chi})~{}=~{}\{X\in G_{k-1}(V)~{}\colon~{}\mbox{either}~{}X\cap V_{1}\neq 0~{}\mbox{or}~{}\pi(X)\in R^{\uparrow}(H_{0})\}.$** 4. (4)

When $k=3$, the points $[p]\not\in[V_{1}]$ have degree $\delta(p)=\delta_{0}(\pi(p))+n-n_{0}$, where $\delta_{0}(\pi(p))$ is the degree of $[\pi(p)]$ with respect to $H_{0}$. The points $p\in[V_{1}]$ have degree $n-1$.

We call $H_{\chi}$ the trivial extension of $H_{0}$ centered at $V_{1}$ (also extension of $H_{0}$ by $V_{1}$, for short) and we denote it by the symbol $\mathrm{Ext}_{V_{1}}(H_{0}).$ When $k=n_{0}$ we have $H_{0}=\emptyset$; we shall call $\mathrm{Ext}_{V_{1}}(\emptyset)$ the trivial hyperplane centered at $V_{1}$. In this case Theorem 3.1 can also be rephrased as follows, with no direct mention of $H_{0}$.

Proposition 3.2** ([ILP17]).**

Let $H=\mathrm{Ext}_{V_{1}}(\emptyset)$ be a trivial hyperplane of $\mathcal{G}_{k}(V)$. Then

[TABLE]

Moreover, $R_{\downarrow}(H)~{}=~{}[V_{1}]$, $R^{\uparrow}(H)~{}=~{}\{X\in\mathcal{G}_{k-1}(V)~{}\colon~{}X\cap V_{1}\neq 0\}$ and, for $X\in\mathcal{G}_{k-2}(V)$, if $X\cap V_{1}\neq 0$ then $R_{X}(H)=[V/X]$, otherwise $R_{X}(H)=[(V_{1}+X)/X]$.

By construction, the lower radical of a trivial extension is never empty. The following theorem shows that the converse is also true, namely if $R_{\downarrow}(H)\neq\emptyset$ then $H$ is a trivial extension, possibly a trivial hyperplane.

If $S$ is a subspace of $V$ with $\dim(S)>k$, denote by $H(S):=\mathcal{G}_{k}(S)\cap H$ the hyperplane of $\mathcal{G}_{k}(S)$ induced by $H.$

Theorem 3.3** ([ILP17]).**

Suppose $R_{\downarrow}(H)\neq\emptyset$ and let $S,S^{\prime}$ be complements in $V$ of the subspace $R<V$ such that $[R]=R_{\downarrow}(H)$. Then

1. (1)

$H~{}=~{}\mathrm{Ext}_{R}(H(S))$; 2. (2)

$H(S)\cong H(S^{\prime})$; 3. (3)

$R_{\downarrow}(H(S))=\emptyset$.

Each hyperplane $H$ of $\mathcal{G}_{k}(V)$ defined by a $k$-linear alternating form $h$ with $\mathrm{rank}\,(h)<\dim V$ is clearly a trivial extension of a hyperplane $H^{\prime}$ of $\mathcal{G}_{k}(V^{\prime})$ with $\dim V^{\prime}=\mathrm{rank}\,(h)$, since $V=\mathrm{Rad}\,(h)\oplus V^{\prime}.$

3.2 Expansions

Let $V_{0}$ be a hyperplane of $V$ and $H_{0}$ a given hyperplane of $\mathcal{G}_{k-1}(V_{0})$. Assume $k\geq 3$; hence $V$ has dimension $n\geq 4$. Put

[TABLE]

Theorem 3.4** ([ILP17]).**

The set $\mathrm{Exp}(H_{0})$ is a hyperplane of $\mathcal{G}_{k}(V)$. Moreover,

1. (1)

$R_{\downarrow}(\mathrm{Exp}(H_{0}))~{}=~{}R_{\downarrow}(H_{0})$. 2. (2)

$R^{\uparrow}(\mathrm{Exp}(H_{0}))~{}=~{}H_{0}\cup\{X\in\mathcal{G}_{k-1}(V)\setminus\mathcal{G}_{k-1}(V_{0})~{}\colon~{}X\cap V_{0}\in R^{\uparrow}(H_{0})\}$. 3. (3)

For $X\in\mathcal{G}_{k-2}(V)$, if $X\subseteq V_{0}$ with $X\in R^{\uparrow}(H_{0})$ then $R_{X}(\mathrm{Exp}(H_{0}))={\mathcal{S}}_{X}(\mathrm{Exp}(H_{0}))=(X)G_{k-1}$ (the latter being computed in $V$). If $X\subseteq V_{0}$ but $X\not\in R^{\uparrow}(H_{0})$ then $R_{X}(\mathrm{Exp}(H_{0}))=(X)H_{0}$ (a subspace of $\mathrm{PG}(V_{0}/X)$). Finally, if $X\not\subseteq V_{0}$, then $R_{X}(\mathrm{Exp}(H_{0}))~{}=~{}\{\langle x,Y\rangle~{}\colon~{}Y\in R_{X\cap V_{0}}(H_{0})\}$ for a given $x\in X\setminus V_{0}$, no matter which.

We call $\mathrm{Exp}(H_{0})$ the expansion of $H_{0}$. A form $h:\bigwedge^{k}V\rightarrow\mathbb{K}$ associated to $\mathrm{Exp}(H_{0})$ can be constructed as follows. Suppose $h_{0}:\bigwedge^{k-1}V_{0}\rightarrow\mathbb{K}$ is the $(k-1)$-alternating linear form defining $H_{0}$. Suppose $V_{0}=\langle e_{1},\ldots,e_{n-1}\rangle$ where $E=(e_{i})_{i=1}^{n}$ is the given basis of $V$. Recall that $\{e_{i_{1}}\wedge\cdots\wedge e_{i_{k}}~{}\colon~{}1\leq i_{1}<\ldots<i_{k}\leq n\}$ is a basis of $\bigwedge^{k}V$. Put

[TABLE]

and extend it by linearity. It is easy to check that the form $h$ defines $\mathrm{Exp}(H_{0})$.

We now recall some properties linking expansions and trivial extensions which might be of use in investigating the geometries involved.

Theorem 3.5** ([ILP17]).**

Let $H_{0}$ be a hyperplane of $\mathcal{G}_{k}(V_{0})$; then

$R_{\downarrow}(\mathrm{Exp}(H_{0}))=\emptyset$* if and only if $R_{\downarrow}(H_{0})=\emptyset$;* 2. 2.

denote by $S_{0}$ a complement of $R_{0}\leq V$ such that $[R_{0}]=R_{\downarrow}(H_{0})$; then, $\mathrm{Exp}(H_{0})=\mathrm{Ext}_{R_{0}}(\mathrm{Exp}(H_{0}(S_{0})))$ where $H_{0}(S_{0})$ is the hyperplane induced on $S_{0}$ by $H_{0}$; 3. 3.

if $H_{0}$ is trivial, then $\mathrm{Exp}(H_{0})$ is also trivial with center $R_{\downarrow}(H_{0})$.

3.2.1 Symplectic hyperplanes

Assume now $k=3$ and take $H_{0}$ to be a hyperplane of $\mathcal{G}_{2}(V_{0})$ (hence defined by a bilinear alternating form of $V_{0}$). The point-line geometry ${\mathcal{S}}(H_{0})=(G_{1}(V_{0}),H_{0})$ having as points all points of $[V_{0}]$ and as lines all elements in $H_{0}$, is a polar space of symplectic type. The upper and lower radical of $H_{0}$ are mutually equal and coincide with the radical $R(H_{0})$ of ${\mathcal{S}}(H_{0})$.

First suppose that ${\mathcal{S}}(H_{0})$ is non-degenerate. Then $n-1$ is even, whence $n\geq 5$. Claims (1) and (2) of Theorem 3.4 imply that $R_{\downarrow}(\mathrm{Exp}(H_{0}))=\emptyset$ and $R^{\uparrow}(\mathrm{Exp}(H_{0}))=H_{0}$; thus the geometry of poles $\mathcal{P}(\mathrm{Exp}(H_{0}))$ of $\mathrm{Exp}(H_{0})$ coincides precisely with the symplectic polar space ${\mathcal{S}}(H_{0}).$ In particular, the points of $[V]\setminus[V_{0}]$ are smooth while those of $[V_{0}]$ are poles of degree $1$. Motivated by the above remark we call $\mathrm{Exp}(H_{0})$ a symplectic hyperplane whenever $R(H_{0})=\emptyset.$

Assume now that ${\mathcal{S}}(H_{0})$ is degenerate, i.e. $\mathrm{Rad}\,({\mathcal{S}}(H_{0}))\neq 0$. In this case, $R_{\downarrow}(\mathrm{Exp}(H_{0}))\neq 0$ since $R_{\downarrow}(\mathrm{Exp}(H_{0}))=\mathrm{Rad}\,({\mathcal{S}}(H_{0}))$; so $\mathrm{Exp}(H_{0})$ is either a trivial extension of a symplectic hyperplane by $\mathrm{Rad}\,({\mathcal{S}}(H_{0}))$ (this happens when $\dim(\mathrm{Rad}\,({\mathcal{S}}(H_{0}))<n-3$) or a trivial hyperplane centered at $\mathrm{Rad}\,({\mathcal{S}}(H_{0}))$ (this happens when $\dim(\mathrm{Rad}\,({\mathcal{S}}(H_{0}))=n-3$).

3.3 Block decomposable hyperplanes

The construction of block decomposable hyperplanes can be done for general $k\geq 3$ but we will give the details for the case $k=3.$

Suppose $V=V_{0}\oplus V_{1}$. Any vector $x\in V$ can then be uniquely written as $x=x_{0}+x_{1}$ with $x_{0}\in V_{0}$ and $x_{1}\in V_{1}.$ For $i=0,1$ let $\bar{h}_{i}:\bigwedge^{3}V_{i}\to\mathbb{K}$ be a linear functional defining the hyperplane $H_{i}$ of $\mathcal{G}_{3}(V_{i})$. Consider the extension $h_{i}:\bigwedge^{3}V\rightarrow\mathbb{K}$ of $\bar{h}_{i}$ to $V$ given by

[TABLE]

where $x=x_{0}+x_{1},y=y_{0}+y_{1},z=z_{0}+z_{1}\in V$ and $x_{i},y_{i},z_{i}\in V_{i}.$

Let $h:=h_{0}+h_{1}$ be the trilinear form of $V$ defined by the sum of $h_{0}$ and $h_{1}.$ So,

[TABLE]

Then the hyperplane of $\mathcal{G}_{3}(V)$ defined by $h$ is called a block decomposable hyperplane arising from $H_{0}$ and $H_{1}$ and it will be denoted by $\mathrm{Dec}(H_{0},H_{1}).$

Theorem 3.6**.**

*Let $H:=\mathrm{Dec}(H_{0},H_{1})$ be a block decomposable hyperplane of $\mathcal{G}_{3}(V).$ Then the following hold: *

The poles of $H$ are all the points of $\mathrm{PG}(V_{0}\oplus V_{1})$; 2. 2.

$R^{\uparrow}(H)=\{\ell\in\mathcal{G}_{2}(V):(\pi_{0}(\ell)\in R^{\uparrow}(H_{0})\mbox{ or }\dim(\pi_{0}(\ell))<2)\mbox{ and }(\pi_{1}(\ell)\in R^{\uparrow}(H_{1})\mbox{ or }\dim(\pi_{1}(\ell))<2)\}$* where $\pi_{i}\colon V\rightarrow V_{i}$ is the projection of $V$ onto $V_{i}$ along $V_{j}$ ($j\neq i$, $i=0,1$). Denote by $\varepsilon_{2}\colon\mathcal{G}_{2}(V)\rightarrow\mathrm{PG}(\bigwedge^{2}V)$ the Plücker embedding of the $2$-Grassmannian $\mathcal{G}_{2}$. We have*

[TABLE]

where $V_{0}\wedge V_{1}:=\langle v_{0}\wedge v_{1}:v_{0}\in V_{0},v_{1}\in V_{1}\rangle.$

Proof.

Put $n_{0}=\dim V_{0}$ and $n_{1}=\dim V_{1}$. Let $u\in V$ where $u=u_{0}+u_{1}\in V$, $u_{0}\in V_{0}$ and $u_{1}\in V_{1}.$ Denote by $M_{u}$ the matrix of the bilinear form $\chi_{u}(x,y):=h(u\wedge x\wedge y)$. Then, $M_{u}$ is a block matrix of the form

[TABLE]

where $M_{u_{i}}$ is the matrix representing the form $\chi_{u}^{i}(x,y):=\bar{h}_{i}(u\wedge x\wedge y)$ associated to the hyperplane $H_{i}$ of $\mathcal{G}_{3}(V_{i}).$

For any $x,y,u\in V$ with $x=x_{0}+x_{1}$, $y=y_{0}+y_{1}$, $u=u_{0}+u_{1}$ and $x_{i},y_{i},u_{i}\in V_{i}$, we have by definition of decomposable hyperplane,

[TABLE]

By Corollary 2.2, $\mathrm{rank}\,(M_{u_{i}})\leq n_{i}-1$. So, $\mathrm{rank}\,M_{u}\leq(n_{0}-1)+(n_{1}-1)=n-2$. By Corollary 2.3, $[u]=[u_{0}+u_{1}]$ is a pole.

A line $\ell=\langle x,y\rangle$ is in the upper radical $R^{\uparrow}(H)$ if, and only if, for any choice of $u\in V$ we have $\chi_{u}(x,y)=0$. Since

[TABLE]

where $x_{i},y_{i}\in V_{i}$ and $x=x_{0}+x_{1}$, $y=y_{0}+y_{1}$, we have $\ell\in R^{\uparrow}(H)$ if and only if for all $u_{i}\in V_{i}$ and $i=0,1$, $\chi_{u_{i}}(x_{i},y_{i})=0.$ This holds if and only if $(\pi_{0}(\ell)\in R^{\uparrow}(H_{0})\mbox{ or }\dim(\pi_{0}(\ell))<2)\mbox{ and }(\pi_{1}(\ell)\in R^{\uparrow}(H_{1})\mbox{ or }\dim(\pi_{1}(\ell))<2).$ The first part of claim 2 is proved.

Consider the following subspace of $\mathrm{PG}(\bigwedge^{3}V)$

[TABLE]

where $V_{0}\wedge V_{1}:=\langle v_{0}\wedge v_{1}:v_{0}\in V_{0},v_{1}\in V_{1}\rangle.$

We claim that any point in $\varepsilon_{2}(R^{\uparrow}(H_{0}))+\varepsilon_{2}(R^{\uparrow}(H_{1}))+V_{0}\wedge V_{1}$ is in $\varepsilon_{2}(R^{\uparrow}(H)).$

Since $R^{\uparrow}(H_{i})\subseteq R^{\uparrow}(H)$ (for $i=0,1$) and $\varepsilon_{2}^{-1}((V_{0}\wedge V_{1})\cap\varepsilon_{2}(\mathcal{G}_{2}(V)))\subseteq R^{\uparrow}(H)$, by the first part of claim 2, the inclusion $\mathcal{L}\subseteq\varepsilon_{2}(R^{\uparrow}(H))$ is immediate.

Suppose now $\langle(x_{0}+x_{1})\wedge(y_{0}+y_{1})\rangle\in\varepsilon_{2}(R^{\uparrow}(H))$. We can write (by the first part of claim 2)

[TABLE]

The thesis follows. ∎

We remark that, with some slight abuse of notation, the extension $\mathrm{Ext}_{V_{1}}(H_{0})$ of an hyperplane $H_{0}$ can always be regarded as a special case of a block decomposable hyperplane, where the form $\bar{h}_{1}$ defined over $V_{1}$ is identically null.

The definition given for block decomposable hyperplane arising from two hyperplanes $H_{0}$ and $H_{1}$ can be extended by induction to the definition of block decomposable hyperplane $\mathrm{Dec}(H_{0},\cdots,H_{n-1})$ arising from $n$ hyperplanes $H_{i}$ ($0\leq i\leq n-1$), where $H_{i}$ is a hyperplane of $\mathcal{G}_{3}(V_{i})$ and $V=\oplus_{i}V_{i}.$

In general, given two linear subspaces $V_{0}$, $V_{1}$ of $V$ such that $V=V_{0}\oplus V_{1}$, and given two hyperplanes $H_{0}$ and $H_{1}$ of $\mathcal{G}_{3}(V_{0})$ and $\mathcal{G}_{3}(V_{1})$, there exist several possible hyperplanes $H$ of $\mathcal{G}_{3}(V_{0}\oplus V_{1})$ which are block decomposable and arise from $H_{0}$ and $H_{1}$, namely all of those induced by the forms $h_{\alpha,\beta}:=\alpha h_{0}+\beta h_{1}$ with $\alpha,\beta\in\mathbb{K}\setminus\{0\}$. Even if all these hyperplanes are in general neither equivalent nor nearly equivalent, they turn out to be always geometrically equivalent and their geometry of poles depends only on the geometries of $H_{0}$ and $H_{1}$.

4 Characterization of the geometry of poles

In this section we will prove our Theorems 1, 2 and 3. As in the previous sections, let $E:=(e_{i})_{i=1}^{n}$ be a given basis of $V.$ Let $H$ be a given hyperplane of $\mathcal{G}_{3}(V)$ and $\mathcal{P}(H)=(P(H),R^{\uparrow}(H))$ be the geometry of poles of $H.$ In Section 2.1 we have explained how to algebraically describe the pointset $P(H)$ and the lineset $R^{\uparrow}(H)$ of the geometry of poles associated to $H.$ The main steps to describe $\mathcal{P}(H)$ are the following:

• •

Consider the bilinear form $\chi_{u}$ associated to the trilinear form defining $H$ and write the matrix $M_{u}$ representing $\chi_{u}$. This is done in Table 2 for forms of rank up to $6$ and in Table 3 for forms of rank $7.$ Recall that for $\dim(V)\leq 7$ and $\mathbb{K}$ perfect with cohomological dimension at most $1$, all trilinear forms are classified: they are listed in Table 1.

• •

By Theorem 2.4, we know that the set of poles is either the pointset of $\mathrm{PG}(V)$ or an algebraic variety. If $\dim(V)=6$ all points of $\mathrm{PG}(V)$ are poles, for any hyperplane $H$ of $\mathcal{G}_{3}(V).$ If $\dim(V)=7$, to get the equations describing the variety $P(H)$ we rely on Corollary 2.3, which gives algebraic conditions on $(u_{i})_{i=1}^{n}$ for $[(u_{i})_{i=1}^{n}]$ to be a $H$-pole. In the second column of Table 5 we have written down those equations, according to the type of $H.$

• •

To describe the lines of $\mathcal{P}(H)$ we rely on the last part of Section 2.1. In particular, $\ell:=[x,y]\in R^{\uparrow}(H)$ if and only if the functional $\tilde{h}_{xy}$ described in Equation (2) is the null functional. This immediately reads as some linear equations in the Plücker coordinates $|x,y|_{ij}$ of the line $\ell$. The results of these (straightforward) computations are reported in the third column of Tables 4 for forms of rank at most $6$ and Table 5 for forms of rank $7$.

So, Tables 1, 2, 3, 4, 5 provide an algebraic description of points and lines of $\mathcal{P}(H),$ for any hyperplane $H$ of $\mathcal{G}_{3}(V).$ In the remainder of this section we shall also provide a geometrical description of $\mathcal{P}(H).$

4.1 Hyperplanes arising from forms of rank at most $6$

4.1.1 Hyperplanes of type $T_{1}$

A hyperplane $H$ of $\mathcal{G}_{3}(V)$ of type $T_{1}$ is defined by a trilinear form of rank $3$ equivalent to $h=\underline{123}$, see the first row of Table 1.

Suppose $\dim(V)\geq 4$ and let $(e_{i})_{i=1}^{n}$ be a given basis of $V.$ Then $\mathrm{Rad}\,(h)=\langle e_{i}\rangle_{i\geq 4}.$ According to Section 3.1, $H$ is a trivial hyperplane $\mathrm{Ext}_{\mathrm{Rad}\,(h)}(\emptyset)$ centered at $\mathrm{Rad}\,(h).$ By Proposition 3.2, the set of $\mathrm{Ext}_{\mathrm{Rad}\,(h)}(\emptyset)$-poles is the whole pointset of $\mathrm{PG}(V)$ and the lines of the geometry of poles, i.e. the elements in the upper radical $R^{\uparrow}(\mathrm{Ext}_{\mathrm{Rad}\,(h)}(\emptyset))$, are those lines of $\mathrm{PG}(V)$ meeting $\mathrm{Rad}\,(h)$ non-trivially. When $n=\dim(V)\leq 6$, this proves part 1 of Theorem 1.

4.1.2 Hyperplanes of type $T_{2}$

A hyperplane $H$ of $\mathcal{G}_{3}(V)$ of type $T_{2}$ is defined by a trilinear form of rank $5$ equivalent to $h=\underline{123}+\underline{145}$, see the second row of Table 1.

Suppose $\dim(V)>5.$ Then $\dim(\mathrm{Rad}\,(h))\geq 1.$ Let $V=\mathrm{Rad}\,(h)\oplus V^{\prime}.$ By Section 3.1, $H$ is a trivial extension $\mathrm{Ext}_{\mathrm{Rad}\,(h)}(H^{\prime})$ of a hyperplane $H^{\prime}$ of $\mathcal{G}_{3}(V^{\prime}).$ By Theorem 3.3, we can assume without loss of generality $V^{\prime}=\langle e_{i}\rangle_{i=1}^{5}.$ Put $V_{0}:=\langle e_{i}\rangle_{i=2}^{5}$ and consider the hyperplane $H_{0}$ of $\mathcal{G}_{2}(V_{0})$ defined by the functional $\underline{23}+\underline{45}.$ By Section 3.2, $H^{\prime}$ is the expansion $\mathrm{Exp}(H_{0})$ of $H_{0}.$ By Subsection 3.2.1, since the geometry $\mathcal{S}(H_{0})=(G_{1}(V_{0}),H_{0})$ is a non-degenerate symplectic polar space, the geometry of poles of $\mathrm{Exp}(H_{0})$ coincides precisely with $\mathcal{S}(H_{0})$. Hence, if $\dim(V)>5$, $H$ is a trivial extension $\mathrm{Ext}_{\mathrm{Rad}\,(h)}(\mathrm{Exp}(H_{0}))$ of a symplectic hyperplane $\mathrm{Exp}(H_{0}).$ The $H$-poles are all the points of $\mathrm{PG}(V)$ and the lines of the geometry of poles are all the lines $\ell$ of $\mathrm{PG}(V)$ meeting $\mathrm{Rad}\,(h)$ non-trivially or such that $\pi(\ell)\in H_{0}$ where $\pi$ is the projection onto $V_{0}$ along $\mathrm{Rad}\,(h).$

If $\dim(V)=5$ then $H$ is the expansion $\mathrm{Exp}(H_{0})$ of a non-degenerate symplectic polar space $\mathcal{S}(H_{0})$ defined by the functional $\underline{23}+\underline{45}$ in $V_{0}=\langle e_{i}\rangle_{i=2}^{5}$. By Subsection 3.2.1, the geometry of poles of $H$ coincides with the symplectic polar space $\mathcal{S}(H_{0}).$

Part 2 of Theorem 1 is proved.

4.1.3 Hyperplanes of type $T_{3}$

A hyperplane $H$ of $\mathcal{G}_{3}(V)$ of type $T_{3}$ is defined by a trilinear form of rank $6$ equivalent to $h=\underline{123}+\underline{456}$, see the third row of Table 1.

Suppose $\dim(V)=6$. Let $(e_{i})_{i=1}^{6}$ be a given basis of $V$. Put $V_{0}=\langle e_{1},e_{2},e_{3}\rangle$ and $V_{1}=\langle e_{4},e_{5},e_{6}\rangle$. Clearly, $V=V_{0}\oplus V_{1}.$ For $i=0,1$, denote by $\bar{h}_{i}:=\underline{(3i+1)(3i+2)(3i+3)}$ the trilinear form induced by the restriction of $h$ to $\bigwedge^{3}V_{i}.$ By Section 3.3, $H$ is a decomposable hyperplane $\mathrm{Dec}(H_{0},H_{1})$ of $\mathcal{G}_{3}(V)$ arising from the hyperplanes $H_{i}$, $i=0,1$, of $\mathcal{G}_{3}(V_{i})$ defined by the forms $\bar{h}_{i}.$ Since $R^{\uparrow}(H_{i})=\emptyset$, by Theorem 3.6, all points of $\mathrm{PG}(V)$ are elements of the geometry of poles $\mathcal{P}(H)$ and the lines of $\mathcal{P}(H)$ are exactly those lines of $\mathrm{PG}(V)$ intersecting both $\mathrm{PG}(V_{0})$ and $\mathrm{PG}(V_{1}).$ This proves part 3 of Theorem 1.

Suppose $\dim(V)>6$. Let $(e_{i})_{i\geq 1}$ be a given basis of $V$. Then $\mathrm{Rad}\,(h)=\langle e_{i}\rangle_{i\geq 7}.$ By last part of Section 3.1, $H$ is a trivial extension $\mathrm{Ext}_{\mathrm{Rad}\,(h)}(\mathrm{Dec}(H_{0},H_{1}))$ of a decomposable hyperplane $\mathrm{Dec}(H_{0},H_{1})$ of $\mathcal{G}_{3}(V^{\prime})$ where $V=\mathrm{Rad}\,(h)\oplus V^{\prime}$ and $H_{i}$, $i=0,1$, are the hyperplanes of $\mathcal{G}_{3}(V_{i})$, $V_{i}=\langle e_{3i+1},e_{3i+2},e_{3i+3}\rangle$, $V^{\prime}=V_{1}\oplus V_{2}$, defined by $\underline{(3i+1)(3i+2)(3i+3)}$.

4.1.4 Hyperplanes of type $T_{4}$

A hyperplane $H$ of $\mathcal{G}_{3}(V)$ of type $T_{4}$ is defined by a trilinear form of rank $6$ equivalent to $h=\underline{162}+\underline{243}+\underline{135}$, see the fourth row of Table 1.

Suppose $\dim(V)=6.$ For any $u\in V$, $\mathrm{rank}\,(M_{u})\leq 4$ (see Table 2 for the description of the matrix $M_{u}$). If $(e_{i})_{i=1}^{6}$ is a basis of $V$ and $V=V_{0}\oplus V_{1}$ with $V_{0}=\langle e_{1},e_{2},e_{3}\rangle$ and $V_{1}=\langle e_{4},e_{5},e_{6}\rangle$, by a direct computation we have that the elements of $V_{1}$ are poles of degree $3$ while all remaining poles have degree $1$.

Let $\ell=[u,v]$ be a line of $\mathrm{PG}(V)$. By the forth row of Table 4, $\ell\in R^{\uparrow}(H)$ if and only if its Plücker coordinates satisfy $6$ linear equations. More explicitly, we have that $\ell=[u,v]\in R^{\uparrow}(H)$ if and only if $u=\bar{u}_{0}+\bar{u}_{1}$ and $v=\omega(\bar{u}_{0})\in V_{1}$ with $\bar{u}_{0}\in V_{0}$, $\bar{u}_{1}\in V_{1}$ and $\omega\colon V\rightarrow V$, $\omega(u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})=(u_{4},u_{5},u_{6},u_{1},u_{2},u_{3}).$ Note that $\omega$ interchanges $V_{0}$ and $V_{1}$. Indeed, if $u=\bar{u}_{0}+\bar{u}_{1}$ and $v=\bar{v}_{0}+\bar{v}_{1}$ with $\bar{u}_{0},\bar{v}_{0}\in V_{0}$ and $\bar{u}_{1},\bar{v}_{1}\in V_{1}$, the Plücker coordinates of the line $\ell=[u,v]$, satisfy the equations $|u,v|_{12}=0,|u,v|_{13}=0,|u,v|_{23}=0$ if and only if $\bar{v}_{0}=\lambda\bar{u}_{0}$ with $\lambda\in\mathbb{K}.$ Hence $\ell=[u,v]=[u,v-\lambda u]=[\bar{u}_{0}+\bar{u}_{1},\bar{v}_{1}^{\prime}]$ with $\bar{v}_{1}^{\prime}=\bar{v}_{1}-\lambda\bar{u}_{1}\in V_{1}.$

The remaining three equations $|x,y|_{26}-|x,y|_{35}=0,|x,y|_{16}-|x,y|_{34}=0,|x,y|_{24}-|x,y|_{15}=0$ are satisfied by the Plücker coordinates of $\ell=[u,v]=[\bar{u}_{0}+\bar{u}_{1},\bar{v}_{1}^{\prime}]$ if and only if either $\bar{u}_{0}$ is the null vector and in this case $\ell=[\bar{u}_{1},\bar{v}_{1}^{\prime}]\subset V_{1}$ or $[\bar{v}_{1}^{\prime}]=[(0,0,0,u_{1},u_{2},u_{3})]$ where $[u]=[(u_{1},u_{2},u_{3},u_{4},u_{5},u_{6})].$

This proves part 4 of Theorem 1.

Suppose $\dim(V)>6$. Let $(e_{i})_{i\geq 1}$ be a given basis of $V$. Then $\mathrm{Rad}\,(h)=\langle e_{i}\rangle_{i\geq 7}.$ By last part of Section 3.1, $H$ is a trivial extension $\mathrm{Ext}_{\mathrm{Rad}\,(h)}(H^{\prime})$ of a hyperplane $H^{\prime}$ of $\mathcal{G}_{3}(V^{\prime})$ where $V=\mathrm{Rad}\,(h)\oplus V^{\prime}$, $V^{\prime}=\langle e_{i}\rangle_{i=1}^{6}$ and $H^{\prime}$ is defined by a trilinear form equivalent to $\underline{162}+\underline{243}+\underline{135}.$ A description of the geometry of poles of $H$ follows from Theorem 3.1 and the already done case of hyperplanes of type $T_{4}$ of $\mathcal{G}_{3}(V)$ for $\dim(V)=6.$

4.1.5 Hyperplanes of type $T_{10,\lambda}^{(i)}$

A hyperplane $H$ of $\mathcal{G}_{3}(V)$ of type $T_{10,\lambda}^{(i)}$ is defined by a trilinear form of rank $6$ as written in row $10$ or $11$ of Table 1, according as $\mathrm{char}\,(\mathbb{K})$ is odd or even. We remark that forms of type $T_{10,\lambda}^{(1)}$ make sense also in even characteristic, provided that the field $\mathbb{K}$ is not perfect.

Let $\dim(V)=6.$ For any $u\in V$, the matrix $M_{u}$ representing the bilinear alternating form $\chi_{u}$ associated to $H$ is written in Table 2. We have that $\mathrm{rank}\,(M_{u})\leq 4$ for every $u\in V$, i.e. every point of $\mathrm{PG}(V)$ is a pole. Actually, we will prove that $\mathrm{rank}\,(M_{u})=4$ for any $u\in V$, i.e. every point of $\mathrm{PG}(V)$ is a pole of degree $1$, equivalently, $R^{\uparrow}(H)$ is a line spread of $\mathrm{PG}(V).$

Consider first a hyperplane of type $T_{10,\lambda}^{(1)}.$ Suppose by way of contradiction that $\mathrm{rank}\,(M_{u})<4,$ i.e. all minors of order 4 vanish. With $u=(u_{1},\ldots,u_{6})$, take the three $4\times 4$ principal minors of $M_{u}$ given by

[TABLE]

By the third column of Table 1 corresponding to $T_{10,\lambda}^{(1)}$, we have that $p_{\lambda}(t)=t^{2}-\lambda$ is an irreducible polynomial in $\mathbb{K}[t].$ Hence the minors mentioned are null if and only if $u_{i}=0$ for all $i=1,\dots,6.$

We argue in a similar way for case $T_{10,\lambda}^{(2)}$ with $\mathrm{char}\,(\mathbb{K})=2$, by choosing the minors of $M_{u}$ given by

[TABLE]

By the third column of Table 1 corresponding to $T_{10,\lambda}^{(2)}$, we have that $p_{\lambda}(t):=t^{2}+\lambda t+1$ is irreducible in $\mathbb{K}[t]$; hence $\mathrm{rank}\,M_{u}=4$ unless $u=(0,\ldots,0)$. So $R^{\uparrow}(H)$ is a line-spread of $\mathrm{PG}(V)$.

Lemma 4.1**.**

Let $H$ be a hyperplane of $\mathcal{G}_{3}(V)$ with $\dim V=6$ whose upper radical $R^{\uparrow}(H)$ is a line-spread of $\mathrm{PG}(V)$. Then $R^{\uparrow}(H)$ is a Desarguesian line-spread of $\mathrm{PG}(V)$.

Proof.

For simplicity of notation, denote by $\mathcal{S}$ the line-spread of $\mathrm{PG}(V)$ induced by $H.$ Then, by duality, $H$ induces also a line spread $\mathcal{S}^{*}$ in the dual space $\mathrm{PG}(V^{*})$, where $V^{*}$ is the dual of $V$. In particular, a $4$-dimensional vector space $\Sigma$ is in $\mathcal{S}^{*}$ if and only if all planes contained in $\Sigma$ are elements of $H$. We will prove that $\mathcal{S}$ is a normal spread, i.e. given any two distinct elements $\ell_{1},\ell_{2}\in\mathcal{S}$ and $\Sigma=\ell_{1}+\ell_{2}$, the set $\mathcal{S}_{\Sigma}:=\{\ell\in\mathcal{S}:\ell\subseteq\Sigma\}$ is a line-spread of $\Sigma$.

Let $\pi$ be a plane contained in $\Sigma$. Then, $\pi\cap\ell_{1}\neq\emptyset\neq\pi\cap\ell_{2}$ and we can write $\pi=[p_{1},p_{2},q_{1}+q_{2}]$ with $p_{1},q_{1}\in\ell_{1}$, $p_{2},q_{2}\in\ell_{2}$ suitably chosen. Denote by $h$ a form defining the hyperplane $H$. Then $h(p_{1}\wedge p_{2}\wedge(q_{1}+q_{2}))=h(p_{1}\wedge p_{2}\wedge q_{1})+h(p_{1}\wedge p_{2}\wedge q_{2})=0$ as $h$ is identically zero on all planes through either $\ell_{1}$ or $\ell_{2}$. Hence $\pi\in H$ and, consequently, $\Sigma\in\mathcal{S}^{*}$.

So the $3$-dimensional projective space spanned by any two elements of $\mathcal{S}$ is in $\mathcal{S}^{*}$; by duality, the intersection of any two elements of $\mathcal{S}^{*}$ is in $\mathcal{S}$. Take now $\Sigma\in\mathcal{S}^{*}$ and $p\in\Sigma$; denote by $\ell_{p}$ the unique line of $\mathcal{S}$ with $p\in\ell_{p}$. Let $\ell^{\prime}\in\mathcal{S}$ such that $\ell^{\prime}$ is not contained in $\Sigma$. Then, $\Sigma^{\prime}=\ell_{p}+\ell^{\prime}\in\mathcal{S}^{*}$ and, by the argument above, $\Sigma^{\prime}\cap\Sigma\in\mathcal{S}$. Since $p\in\Sigma^{\prime}\cap\Sigma$, it follows that $\Sigma^{\prime}\cap\Sigma=\ell_{p}$. This for all points $p\in\Sigma$; so $\mathcal{S}_{\Sigma}$ is a spread of $\Sigma$. Hence $\mathcal{S}$ is a normal spread and by [BC72, Theorem 2] $\mathcal{S}$ is a Desarguesian line spread of $\mathrm{PG}(V).$ ∎

Part 5 of Theorem 1 is now proved.

Proof of Theorem 2. By Lemma 4.1, line-spreads of $\mathrm{PG}(6,V)$ induced by hyperplanes of $\mathcal{G}_{3}(V)$ are Desarguesian. As Desarguesian line-spreads are coordinatized over division rings, there must exist a division ring ${\mathbb{D}}$ having dimension $2$ over $\mathbb{K}$ (see [BB66]) with either ${\mathbb{D}}$ commutative or $\mathbb{K}$ being the center of ${\mathbb{D}}$. We show that ${\mathbb{D}}$ is commutative. Take $a\in{\mathbb{D}}\setminus\mathbb{K}$. Then the algebraic extension $\mathbb{K}(a)$ is a proper field extension of $\mathbb{K}$ contained in ${\mathbb{D}}$; so $2=[{\mathbb{D}}:\mathbb{K}]\geq[\mathbb{K}(a):\mathbb{K}]\geq 2$. It follows that ${\mathbb{D}}=\mathbb{K}(a)$ and ${\mathbb{D}}$ is a field which is an algebraic extension of degree $2$ of $\mathbb{K}$. So, for ${\mathbb{D}}$ to exist, $\mathbb{K}$ must not be quadratically closed.                                                                                           $\Box$

Remark 1*.*

The commutativity of ${\mathbb{D}}$ in the proof of Theorem 2 is also a consequence of the Artin-Wedderburn theorem. We have provided a short argument.

Suppose $\dim(V)>6$. Let $(e_{i})_{i\geq 1}$ be a given basis of $V$. Then $\mathrm{Rad}\,(h)=\langle e_{i}\rangle_{i\geq 7}.$ By last part of Section 3.1, $H$ is a trivial extension $\mathrm{Ext}_{\mathrm{Rad}\,(h)}(H^{\prime})$ of a hyperplane $H^{\prime}$ of $\mathcal{G}_{3}(V^{\prime})$ where $V=\mathrm{Rad}\,(h)\oplus V^{\prime}$, $V^{\prime}=\langle e_{i}\rangle_{i=1}^{6}$ and $H^{\prime}$ is defined by a trilinear form of type $T_{10,\lambda}^{(i)}.$ A description of the geometry of poles of $H$ follows from Theorem 3.1 and the already done case of hyperplanes of type $T_{10,\lambda}^{(i)}$ of $\mathcal{G}_{3}(V)$ with $\dim(V)=6.$

4.2 Hyperplanes arising from forms of rank $7$

Throughout this section let $[u]=[(u_{i})_{i=1}^{7}]$.

4.2.1 Hyperplanes of type $T_{5}$

A hyperplane $H$ of $\mathcal{G}_{3}(V)$ of type $T_{5}$ is defined by a trilinear form of rank $7$ equivalent to $h=\underline{123}+\underline{456}+\underline{147}$, see the fifth row of Table 1.

Suppose $\dim(V)=7.$ Straightforward computations shows that $\mathrm{rank}\,(M_{u})\leq 4$ (see Table 3 for the description of the matrix $M_{u}$) if and only if $u_{1}=0$ or $u_{4}=0$ and $\mathrm{rank}\,(M_{u})=2$ if and only if $u_{1}=u_{4}=u_{5}=u_{6}=0$ or $u_{1}=u_{2}=u_{3}=u_{4}=0.$

Let $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ the hyperplanes of $\mathrm{PG}(V)$ with equations respectively $u_{1}=0$ and $u_{4}=0$. Denoted by $P(H)$ the set of poles of $H$, we then have $P(H)=\mathcal{S}_{1}\cup\mathcal{S}_{2}.$ A point $[u]$ has degree $4$ if and only if $[u]\in A_{1}\cup A_{2}$ where $A_{1}$ is the plane of $\mathrm{PG}(V)$ of equation $u_{1}=u_{4}=u_{5}=u_{6}=0$ and $A_{2}$ is the plane of $\mathrm{PG}(V)$ of equation $u_{1}=u_{2}=u_{3}=u_{4}=0.$

Take $[u]=[(0,u_{2},\ldots,u_{7})]\in\mathcal{S}_{1}$. In this case, by the equations of Table 5, a line $\ell:=[u,v]$ of $\mathrm{PG}(V)$ through $[u]$ is in $R^{\uparrow}(H)$ if and only if $\ell\subseteq\mathcal{S}_{1}$, intersects $A_{1}$ non-trivially and it is totally isotropic for the non-degenerate bilinear alternating form of $\mathcal{S}_{1}$ defined by $\beta(u,v)=u_{2}v_{3}-u_{3}v_{2}+u_{4}v_{7}-u_{7}v_{4}+u_{5}v_{6}-u_{6}v_{5}.$ A similar argument shows that if $[u]$ is taken in $\mathcal{S}_{2}$, then any line of $R^{\uparrow}(H)$ through it meets $A_{2}$ and it is totally isotropic for the non-degenerate bilinear alternating form of $\mathcal{S}_{2}$ defined by $\beta(u,v)=u_{1}v_{7}-u_{7}v_{1}+u_{2}v_{3}-u_{3}v_{2}+u_{5}v_{6}-u_{6}v_{5}.$

This proves part 1 of Theorem 3.

4.2.2 Hyperplanes of type $T_{6}$

A hyperplane $H$ of $\mathcal{G}_{3}(V)$ of type $T_{6}$ is defined by a trilinear form of rank $7$ equivalent to $h=\underline{152}+\underline{174}+\underline{163}+\underline{243}$, see the sixth row of Table 1. Suppose $\dim V=7$; straightforward computations show that $\mathrm{rank}\,(M_{u})\leq 4$ (see Table 3 for the description of the matrix $M_{u}$) if and only if $u_{1}=0$. Let $\mathcal{S}$ be the hyperplane of $\mathrm{PG}(V)$ of equation $u_{1}=0$. Then the set of the $H$-poles is precisely the point-set of $\mathcal{S}$. Also, a point $[u]$ has degree $4$ if and only if $[u]\in A$ where $A$ is the plane of equation $u_{1}=u_{2}=u_{3}=u_{4}=0$. Take $u=[(0,u_{2},\ldots,u_{7})]\in\mathcal{S}$. By the equations $|u,v|_{34}=|u,v|_{24}=|u,v|_{23}$, of Table 5, each line of the upper radical must intersect the plane $A$. By the equation $|u,v|_{25}+|u,v|_{36}+|u,v|_{47}=0$, we have that a line $\ell=[u,v]$ is in $R^{\uparrow}(H)$ if and only if $\ell\subseteq\mathcal{S}$, $\ell\cap A\neq 0$ and $\ell$ is totally isotropic for the non-degenerate alternating form $\beta(u,v):=|u,v|_{25}+|u,v|_{36}+|u,v|_{47}=0$. This proves part 2 of Theorem 3.

4.2.3 Hyperplanes of type $T_{7}$

A hyperplane $H$ of $\mathcal{G}_{3}(V)$ of type $T_{7}$ is defined by a trilinear form of rank $7$ equivalent to $h=\underline{146}+\underline{157}+\underline{245}+\underline{367},$ see the seventh row of Table 1. Suppose $\dim V=7$; straightforward computations show that $\mathrm{rank}\,(M_{u})\leq 4$ (see Table 3 for the description of the matrix $M_{u}$) if and only if $u_{5}u_{7}+u_{4}u_{6}=0$. Let $A$ be the plane of $\mathrm{PG}(V)$ of equations $u_{4}=u_{5}=u_{6}=u_{7}=0$ and denote by $\mathcal{Q}$ the hyperbolic quadric of equation $u_{5}u_{7}+u_{4}u_{6}=0$ embedded in the subspace $W$ of equations $u_{1}=u_{2}=u_{3}=0$. The set of $H$-poles is the point-set of the quadratic cone of $\mathrm{PG}(V)$ with vertex $A$ and basis $\mathcal{Q}$. Also, a point $[u]$ has degree $4$ if and only if $[u]\in\mathcal{C}$ where $\mathcal{C}$ is the conic of $A$ with equation $u_{1}^{2}-u_{2}u_{3}=0$. Using the equations of Table 5, it is possible to associate to any $[x]=[(st,t^{2},s^{2},0,0,0,0)]\in\mathcal{C}$ a unique line $\ell_{x}=[(0,0,0,s,0,0,t),(0,0,0,0,s,-t,0)]$ of $\mathcal{Q}$.

With $[x]\in\mathcal{C}$, denote by $\operatorname{Res}_{\langle A,\ell_{x}\rangle}(x)$ the projective geometry whose points are all the $2$-dimensional vector spaces through $x$ in the $5$-dimensional vector space $\langle A,\ell_{x}\rangle$ and whose lines are the $3$-dimensional vector spaces of $\langle A,\ell_{x}\rangle$ through $x$. Note that $\operatorname{Res}_{\langle A,\ell_{x}\rangle}(x)\cong\mathrm{PG}(3,\mathbb{K})$.

Proposition 4.2**.**

For any $[x]\in\mathcal{C}$ there exists a line spread $\mathcal{S}_{x}$ of $\operatorname{Res}_{\langle A,\ell_{x}\rangle}(x)$ such that $\ell\in R^{\uparrow}(H)$ if and only if $\ell\subseteq\langle x,s\rangle$ for some $x\in\mathcal{C}$ and $s\in\mathcal{S}_{x}$.

Proof.

Suppose $x,x^{\prime}\in\mathcal{C}$ with $x\neq x^{\prime}$. Then, $\langle A,\ell_{x}\rangle\cap\langle A,\ell_{x^{\prime}}\rangle=A$. We shall now define $\mathcal{S}_{x}$ as follows

[TABLE]

where $\pi_{p}$ is the unique plane of $\mathrm{PG}(V)$ spanned by all the lines through $p$ in $R^{\uparrow}(H)$. Note that $\delta(p)=2$ and $x\in\pi_{p}\subseteq\langle A,\ell_{x}\rangle$.

Let now $[q]$ be a point in $\pi_{p}$ with $\pi_{p}\in\mathcal{S}_{x}$. If $[q]\not\in[x,p]$, then $[x,q]$ and $[p,q]$ are both lines in the upper radical of $H$ through $[q]$; since $\delta(q)=2$ we have $\pi_{q}=\pi_{p}$. If $[q]\in[x,p]$ we consider a point $[r]\not\in[x,p]$ and apply the same argument to show that $\pi_{q}=\pi_{r}=\pi_{p}$. Hence all lines of $\mathrm{PG}(V)$ in any plane $\pi_{p}\in\mathcal{S}_{x}$ are in the upper radical and for any pole $p\not\in A$, the lines of the upper radical through $p$ are contained in $\pi_{p}$.

Suppose $\pi_{p}$ and $\pi_{q}$ are two lines of $\mathcal{S}_{x}$ with non-trivial intersection, i.e. $\pi_{p}\cap\pi_{q}$ is a line through $x$ not contained in $A$. Any point on this line would have degree at least $3$ — a contradiction.

Finally all lines contained in $A$ are elements of the upper radical and $A$ reads as one line of $\mathcal{S}_{x}$. ∎

This proves part 3 of Theorem 3.

4.2.4 Hyperplanes of type $T_{8}$

A hyperplane $H$ of $\mathcal{G}_{3}(V)$ of type $T_{8}$ is defined by a trilinear form of rank $7$ equivalent to $h=\underline{123}+\underline{145}+\underline{167}$, see the eighth row of Table 1. Suppose $\dim V=7$. Put $V_{0}=\langle e_{i}\rangle_{i=2}^{7}$ and consider the hyperplane $H_{0}$ of $\mathcal{G}_{2}(V_{0})$ define by the functional $\underline{23}+\underline{45}+\underline{67}$. By Subsection 3.2, $H$ is the expansion $\mathrm{Exp}(H_{0})$ of $H_{0}$ and since the geometry $S(H_{0}):=(G_{1}(V_{0}),H_{0})$ is a non-degenerate symplectic polar space, the geometry of poles of $\mathrm{Exp}(H_{0})$ coincides with $S(H_{0})$.

This proves part 4 of Theorem 3.

4.2.5 Hyperplanes of type $T_{9}$ and $T_{12,\mu}$

Hyperplanes $H$ of $\mathcal{G}_{3}(V)$ of types either $T_{9}$ or $T_{12,\mu}$ are defined by trilinear forms of rank $7$ nearly equivalent to $h=\underline{123}+\underline{456}+\underline{147}+\underline{257}+\underline{367}$, see rows $9$ and $14$ of Table 1. Suppose $\dim V=7$; straightforward computations show that $\mathrm{rank}\,(M_{u})\leq 4$ (see Table 3 for the description of the matrix $M_{u}$) if and only if $[u]$ is a point of a non-degenerate parabolic quadric $\mathcal{Q}$ of $\mathrm{PG}(V)$. More precisely, each point of $\mathcal{Q}$ has degree $2$.

The equations appearing in Table 5 for cases $T_{9}$ (and $T_{12,\mu}$) are the same as those in [VM98, §2.4.13] for the standard embedding of the Split-Cayley hexagon $H(\mathbb{K})$ in $\mathrm{PG}(6,\mathbb{K})$; see also [Ti59]. Hence, the geometry of the poles of $H$ is precisely a Split-Cayley hexagon. For these reasons, hyperplanes of this type are called hexagonal. This proves part 5 of Theorem 3.

4.2.6 Hyperplanes of type $T_{11,\lambda}^{(i)}$

A hyperplane $H$ of $\mathcal{G}_{3}(V)$ of type $T_{11,\lambda}^{(i)}$ ($i=1,2$) is defined by a trilinear form of rank $7$ as written in row $12$ or $13$ of Table 1, according as $\mathrm{char}\,(\mathbb{K})$ is odd or even. As in the case $T_{10,\lambda}^{(1)}$, we remark that forms of type $T_{11,\lambda}^{(1)}$ may also be considered in even characteristic, provided that the field is not perfect. The geometries of poles arising in both cases $i=1,2$ afford a similar description.

Suppose $\dim V=7$. By the proof of Theorem 2.4, straightforward computations show that $\mathrm{rank}\,(M_{u})\leq 4$ if and only if $\det(M_{u}^{(7)})/u_{7}^{2}=0$ where by $M_{u}^{(7)}$ is the submatrix of $M_{u}$ (see Table 5 for the description of $M_{u}$) obtained by deleting its last row and column.

First consider the case $T_{11,\lambda}^{(1)}$. Then $[u]$ is a pole if and only if its coordinates satisfy the equation $\lambda u_{4}^{2}-u_{1}^{2}=0$. Since the polynomial $p_{\lambda}(x)=x^{2}-\lambda$ is irreducible in $\mathbb{K}$, the points satisfying the above equation have coordinates with $u_{1}=u_{4}=0$. Hence, the set of poles is $\mathcal{S}:=\mathcal{S}_{1}\cap\mathcal{S}_{2}$ where $\mathcal{S}_{1}$ is the hyperplane of $\mathrm{PG}(V)$ of equation $u_{1}=0$ and $\mathcal{S}_{2}$ is the hyperplane of $\mathrm{PG}(V)$ of equation $u_{4}=0$. Considering the $4\times 4$ principal minors of $M_{u}$ given by

[TABLE]

we have that the only point of degree $4$ is $[e_{7}]$.

In the case $T_{11,\lambda}^{(2)}$, then $[u]$ is a pole if and only if its coordinates satisfy the equation $u_{4}^{2}+\lambda u_{1}u_{4}+u_{1}^{2}=0$. Since the polynomial $p_{\lambda}(x)=x^{2}+\lambda x+1$ is irreducible in $\mathbb{K}$, the points satisfying the above equation have coordinates with $u_{1}=u_{4}=0$. Hence, the set of poles is $\mathcal{S}:=\mathcal{S}_{1}\cap\mathcal{S}_{2}$ where $\mathcal{S}_{1}$ is the hyperplane of $\mathrm{PG}(V)$ of equation $u_{1}=0$ and $\mathcal{S}_{2}$ is the hyperplane of $\mathrm{PG}(V)$ of equation $u_{4}=0$. Considering the $4\times 4$ principal minors of $M_{u}$ given by

[TABLE]

so we have that the only point of degree $4$ is $[e_{7}]$ as above.

We now provide a geometric description of $R^{\uparrow}(H)$ holding for both $i=1$ and $i=2$. Denote by $\operatorname{Res}_{\mathcal{S}}(e_{7})$ the projective geometry whose points are all the lines of $\mathcal{S}$ through $[e_{7}]$ and whose lines are the planes of $\mathcal{S}$ through $[e_{7}]$. It is well-known that $\operatorname{Res}_{\mathcal{S}}(e_{7})\cong\mathrm{PG}(3,\mathbb{K})$. Consider the following set

[TABLE]

where $\pi_{p}$ is the plane of $\mathrm{PG}(V)$ spanned by the lines in $R^{\uparrow}(H)$.

Proposition 4.3**.**

The set $\mathcal{F}$ is a line-spread of $\operatorname{Res}_{\mathcal{S}}(e_{7})$ and

[TABLE]

Proof.

Take $[p]\in\mathcal{S}\setminus[e_{7}]$. Since $[p]$ a pole of degree $2$, then there exists a plane $\pi_{p}$ with $[p]\in\pi_{p}$ spanned by lines of $R^{\uparrow}(H)$. Note that $[p,e_{7}]\in R^{\uparrow}(H)$; so $[e_{7}]\in\pi_{p}$. Any line $\ell\subseteq\pi_{p}$ is in the upper radical of $H$. Indeed, let now $[q]$ be a point in $\pi_{p}$ with $\pi_{p}\in\mathcal{F}$. If $[q]\not\in[x,p]$, then $[x,q]$ and $[p,q]$ are both lines in the upper radical of $H$ through $[q]$; since $\delta(q)=2$ we have $\pi_{q}=\pi_{p}$. If $[q]\in[x,p]$ we consider a point $[r]\not\in[x,p]$ and apply the same argument to show that $\pi_{q}=\pi_{r}=\pi_{p}$. Hence all lines of $\mathrm{PG}(V)$ in any plane $\pi_{p}\in\mathcal{F}$ are in the upper radical of $H$.

Suppose now $\pi_{p},\pi_{q}\in\mathcal{F}$ and $\pi_{p}\cap\pi_{q}=r$ where $r$ is a line through $[e_{7}]$. Then any point on $r$ has degree at least $3$, a contradiction. We have proved that the set $\mathcal{F}$ is a line-spread of $\operatorname{Res}_{\mathcal{S}}(e_{7})$. The characterization of the upper radical is now straightforward. ∎

This proves part 6 of Theorem 3.

A Tables

According to the clauses assumed on $\lambda$, types $T^{(r)}_{s,\lambda}$ ($r\in\{1,2\}$, $s\in\{10,11\}$) can be considered only when $\mathbb{K}$ is not quadratically closed. Moreover, when $\lambda\neq\lambda^{\prime}$ the types $T^{(r)}_{s,\lambda}$ and $T^{(r)}_{s,\lambda^{\prime}}$ are different up to linear and near equivalence, even if they might describe geometrically equivalent forms.

It follows from Revoy [Revoy79] and Cohen and Helminck [CH88] that two functionals of types $T_{i}$ and $T_{j}$ with $1\leq i<j\leq 9$ are never nearly equivalent; a functional of type $T_{i}$ with $i\leq 9$ is never nearly equivalent to a functional of type $T^{(r)}_{s,\lambda}$; two functionals of type $T^{(r)}_{s,\lambda}$ and $T^{(r^{\prime})}_{s^{\prime},\lambda^{\prime}}$ with $(r,s)\neq(r^{\prime},s^{\prime})$ are never nearly equivalent while two functionals of type $T^{(r)}_{s,\lambda}$ and $T^{(r)}_{s,\lambda^{\prime}}$ are nearly equivalent if and only if, denoted by $\mu$ and $\mu^{\prime}$ respectively a root of $p_{\lambda}(t)$ and a root of $p_{\lambda^{\prime}}(t)$ in the algebraic closure of $\mathbb{K}$, we have $\mathbb{K}(\mu)=\mathbb{K}(\mu^{\prime})$ (see the fourth column of Table 1 for the definition of $p_{\lambda}(t)$). The forms $T_{9}$ and $T_{12,\mu}$ are not linearly equivalent; however they are, by construction, nearly equivalent.

Also, the forms $T_{10,\lambda}^{(i)}$ and $T_{10,\lambda^{\prime}}^{(i)}$ as well as $T_{11,\lambda}^{(i)}$ and $T_{11,\lambda^{\prime}}^{(i)}$ are not in general nearly-equivalent however they are geometrically equivalent. In particular both $T_{10,\lambda}^{(1)}$ and $T_{10,\lambda^{\prime}}^{(2)}$ induce a Desarguesian spread on $\mathrm{PG}(V)$. Note that the forms $T_{10,\lambda}^{(1)}$ exist only if $\mathrm{char}\,(\mathbb{K})\neq 2$ or if $\mathrm{char}\,(\mathbb{K})=2$ and $\mathbb{K}$ is not perfect. The forms of type $T_{10,\lambda}^{(2)}$ are equivalent to form of type $T_{10,\lambda}^{(1)}$ if $\mathrm{char}\,(\mathbb{K})\neq 2$; however, they are not equivalent if $\mathrm{char}\,(\mathbb{K})=2$.