2·1. Normal rational curve and osculating spaces

An arc in $\mathrm{PG}(k-1,q)$ is a set of points with the property that any k of them span the entire space. One of the most prominent examples of an arc in a projective space is the normal rational curve.

Definition 2·1. A normal rational curve $\mathcal{N}_{k,q}$ over ${\mathbb F}_q$ is the set of points in $\mathrm{PG}(k-1,q)$ defined as

(2·1) \begin{equation} \{\langle (g_0(u,t),g_1(u,t),\ldots,g_{k-1}(u,t)) \rangle_{{\mathbb F}_q} \,:\, u,t \in {\mathbb F}_q, \ (u,t) \neq (0,0)\},\end{equation}

where $g_0(x,y),g_1(x,y),\ldots,g_{k-1}(x,y) \in {\mathbb F}_q[x,y]$ form a basis for the space of homogeneous polynomials in ${\mathbb F}_q[x,y]$ of degree $k-1$ .

A normal rational curve in $\mathrm{PG}(k-1,q)$ consists of $q+1$ points and, when $q \geq k-1$ , it forms an arc of $\mathrm{PG}(k-1,q)$ , see, e.g., [ Reference Hirschfeld and Thas26 , theorem 6·30]. If the condition $q \geq k-1$ is not satisfied, then the normal rational curve contains fewer than k points. Consequently, $\mathcal{N}_{k,q}$ spans at most a hyperplane of $\mathrm{PG}(k-1,q)$ and in this case, $\mathcal{N}_{k,q}$ defines an arc in the subspace spanned by $\mathcal{N}_{k,q}$ itself. For further basic properties of the normal rational curve, we refer the reader to [ Reference Ball and Lavrauw7 , Reference Harris22 , Reference Hirschfeld and Thas26 ].

In view of (2·1), the normal rational curve $\mathcal{N}_{k,q}$ , can be regarded as the image in $\mathrm{PG}(k-1,q)$ of the projective line $\mathrm{PG}(1,q)$ under the Veronese embedding

(2·2) \begin{equation} \nu_{k,q}\colon \langle (u,t) \rangle_{{\mathbb F}_q} \longmapsto \langle (g_0(u,t),g_1(u,t),\ldots,g_{k-1}(u,t)) \rangle_{{\mathbb F}_q},\end{equation}

where

\begin{equation*}g_i(x,y)=x^{k-1-i}y^{i} \in {\mathbb F}_q[x,y],\end{equation*}

for every $i\in\{0,\ldots,k-1\}$ . We will also denote the image $\nu_{k,q}\big(\langle(u,t)\rangle_{{\mathbb F}_q}\big)$ simply by $\nu_{k,q}(u,t)$ whenever no confusion can arise. For $j\ge 1$ , let

\begin{equation*}\frac{\partial^{\,j} g_i}{\partial x^{\,j}}\quad\text{and}\quad\frac{\partial^{\,j} g_i}{\partial y^{\,j}}\end{equation*}

denote the usual jth partial derivatives of the polynomials $g_i(x,y)$ . When $\mathrm{char}({\mathbb F}_q)=p \gt j$ and $k-1 \gt j$ , not all coefficients of the partial derivatives are zero. We can then define the j-th derivative of the parametrisation with respect to x and y by

\begin{equation*}\frac{\partial^{\,j}\nu_{k,q}}{\partial x^{\,j}}(x,y)\,:\!=\,\big\langle\big(\tfrac{\partial^{\,j}}{\partial x^{\,j}} g_0(x,y),\dots,\tfrac{\partial^{\,j}}{\partial x^{\,j}} g_{k-1}(x,y)\big)\big\rangle_{{\mathbb F}_q}\end{equation*}

and

\begin{equation*}\frac{\partial^{\,j}\nu_{k,q}}{\partial y^{\,j}}(x,y)\,:\!=\,\big\langle\big(\tfrac{\partial^{\,j}}{\partial y^{\,j}} g_0(x,y),\dots,\tfrac{\partial^{\,j}}{\partial y^{\,j}} g_{k-1}(x,y)\big)\big\rangle_{{\mathbb F}_q},\end{equation*}

respectively. Each point $\langle (u,t) \rangle_{{\mathbb F}_q} = \langle (1, {t}/{u}) \rangle_{{\mathbb F}_q}$ , $u \ne 0$ , is determined by ${t}/{u} \in {\mathbb F}_q$ . Therefore, each point of $\mathrm{PG}(1, q)$ is represented by a single element of ${\mathbb F}_q \cup \{\infty\}$ , where $\langle (0,1) \rangle_{{\mathbb F}_q}$ corresponds to $\infty$ . Hence the coordinate functions become $g_i(1,t)=t^i$ , and the curve $\mathcal{N}_{k,q}$ can also be written as

(2·3) \begin{equation} \bigl\{\big\langle \big(1,t,\ldots,t^{k-1}\big) \big\rangle_{{\mathbb F}_q} \,:\, t \in {\mathbb F}_q \,\bigr\} \cup \big\{\langle (0,\ldots,0,1)\big\rangle_{{\mathbb F}_q}\big\}.\end{equation}

Note that the jth derivative with respect to t is

\begin{equation*} \frac{\mathrm d^{\,j}}{\mathrm d t^{\,j}} t^i=i(i-1)\cdots(i-j+1)\,t^{\,i-j} =\prod_{\ell=0}^{j-1}(i-\ell)\,t^{\,i-j}.\end{equation*}

Since $p \gt j$ , no coefficient in this product vanishes in ${\mathbb F}_q$ whenever $i\ge j$ . Hence the jth derivative of the Veronese map $\nu_{k,q}$ at the point $\langle(1,t)\rangle_{{\mathbb F}_q}$ is

(2·4) \begin{align} \frac{\partial^{\,j}}{\partial t^{\,j}}\nu_{k,q}(1,t) & =\Bigg\langle\Bigg(\underbrace{0,\ldots,0}_{j\ \text{zeros}},\prod_{\ell=0}^{j-1}(j-\ell),\prod_{\ell=0}^{j-1}(j+1-\ell)\,t,\prod_{\ell=0}^{j-1}(j+2-\ell)\,t^{2},\ldots,\nonumber\\& \quad \ \prod_{\ell=0}^{j-1}(k-1-\ell)\,t^{\,k-1-j}\Bigg)\Bigg\rangle_{{\mathbb F}_q},\end{align}

where the first j entries vanish because the jth derivative of $t^i$ is zero for $i \lt j$ . Therefore, the formulae above describe the derivatives of the Veronese map at every point of the normal rational curve, except at the point $\langle (0,\ldots,0,1) \rangle_{{\mathbb F}_q}$ , that is described below. The jth derivative of the Veronese map at the point $\langle (u, 1) \rangle_{{\mathbb F}_q}$ is

\begin{align*}\frac{\partial^{\,j}}{\partial u^{\,j}}\nu_{k,q}(u,1) & =\Bigg\langle \Bigg(\prod_{\ell=0}^{j-1}(k-1-\ell)\,u^{\,k-1-j},\;\prod_{\ell=0}^{j-1}(k-2-\ell)\,u^{\,k-2-j},\;\dots,\\& \quad\ \prod_{\ell=0}^{j-1}(j-\ell)\,u^{\,0},\;\underbrace{0,\ldots,0}_{j}\Bigg)\Bigg\rangle_{{\mathbb F}_q}.\end{align*}

Evaluating at $u=0$ , the first $k-1-j$ entries vanish. Therefore,

(2·5) \begin{equation} \frac{\partial^{\,j}}{\partial u^{\,j}}\nu_{k,q}(0,1)=\Bigg\langle \Bigg(\underbrace{0,\ldots,0}_{k-j-1},1,\underbrace{0,\ldots,0}_{j}\Bigg) \Bigg\rangle_{{\mathbb F}_q}.\end{equation}

Since $p \gt j$ , in both cases each of the sets of points

\begin{equation*}\left\{\nu_{k,q}(1,t),\;\frac{\partial}{\partial t}\nu_{k,q}(1,t),\;\dots,\;\frac{\partial^{\,j}}{\partial t^{\,j}}\nu_{k,q}(1,t) \right\}\!, \quad \mbox{ with }t \in {\mathbb F}_q,\end{equation*}

and

\begin{equation*} \left\{\nu_{k,q}(0,1),\;\frac{\partial}{\partial u}\nu_{k,q}(0,1),\;\dots,\;\frac{\partial^{\,j}}{\partial u^{\,j}}\nu_{k,q}(0,1) \right\}\end{equation*}

span a $(j-1)$ -dimensional projective space of $\mathrm{PG}(hk-1, q)$ . So, in view of the discussion above, we are now ready to define the osculating spaces of a normal rational curve.

Definition 2·2. Assume that the characteristic of the field is $\mathrm{char}({\mathbb F}_q)=p \gt j$ , for $j\in\{1,\ldots,k-2\}$ and consider the normal rational curve

\begin{equation*}\mathcal{N}_{k,q}=\bigl\{\nu_{k,q}(u,t)\;:\;\langle(u,t)\rangle_{{\mathbb F}_q}\in\mathrm{PG}(1,q)\bigr\} \subseteq \mathrm{PG}(k-1,q).\end{equation*}

For every point $\nu_{k,q}(1,t)$ with $t\in{\mathbb F}_q$ , the osculating space of order j at $\nu_{k,q}(1,t)$ is the j-dimensional projective space

\begin{equation*}\mathrm{Osc}_j(\nu_{k,q}(1,t))\,:\!=\,\bigg\langle\nu_{k,q}(1,t),\frac{\partial}{\partial t}\nu_{k,q}(1,t),\dots,\frac{\partial^{\,j}}{\partial t^{\,j}}\nu_{k,q}(1,t)\bigg\rangle,\end{equation*}

whereas the osculating space of order j at $\nu_{k,q}(0,1)$ is the j-dimensional projective space

\begin{equation*}\mathrm{Osc}_j\bigl(\nu_{k,q}(0,1)\bigr)\,:\!=\,\bigg\langle\nu_{k,q}(0,1),\frac{\partial}{\partial u}\nu_{k,q}(0,1),\dots,\frac{\partial^{\,j}}{\partial u^{\,j}}\nu_{k,q}(0,1)\bigg\rangle.\end{equation*}

When $j=1$ , the osculating space of order 1 at a point P is simply the tangent line to the curve at P. Also, for every point P one obtains a natural flag of osculating spaces

\begin{equation*}P \;\subset\; \mathrm{Osc}_1(P) \;\subset\; \mathrm{Osc}_2(P)\;\subset\; \cdots \;\subset\; \mathrm{Osc}_{k-2}(P).\end{equation*}

Finally, with ${\mathcal N}_{k, q}$ as in (2·3), the transformation given by $t \mapsto (at+b)/(ct+d)$ , with $a,b,c,d \in {\mathbb F}_q$ , $ad - bc \ne 0$ induces the transformation

\begin{align*} (1, t, \dots, t^{k-1}) \mapsto & ((at+b)^{k-1}, (at+b)^{k-2}(ct+d), \dots, (at+b)(ct+d)^{k-2},(ct+d)^{k-1}) \\ & = (1, t, \dots, t^{k-1}) T\end{align*}

for a suitable non-singular matrix T. Hence there is a subgroup of $\mathrm{PGL}(k, q)$ isomorphic to $\mathrm{PGL}(2, q)$ which fixes ${\mathcal N}_{k, q}$ , see e.g. [ Reference Hirschfeld and Thas26 , theorem 6·30(vi)]. Moreover, by selecting the projectivities induced by those matrices T with determinant one, we obtain a group isomorphic to $\mathrm{PSL}(2, q)$ stabilising ${\mathcal N}_{k, q}$ .

Observe that H has order $(q^3-q)/d$ , where $d = \gcd(2, q-1)$ , [ Reference Hirschfeld24 , p. 261].

2·2. Desarguesian spreads

An isomorphism from ${\mathcal P}_1$ to ${\mathcal P}_2$ , with ${\mathcal P}_1$ , ${\mathcal P}_2$ both projective spaces of the same dimensions and defined over the same finite field, is a mapping from ${\mathcal P}_1$ to ${\mathcal P}_2$ preserving the incidence of points and lines. If such a map exists, then ${\mathcal P}_1$ and ${\mathcal P}_2$ are said to be isomorphic, denoted by ${\mathcal P}_1 \simeq {\mathcal P}_2$ . If ${\mathcal P}_1 = {\mathcal P}_2$ , then an isomorphism from ${\mathcal P}_1$ to itself is called a collineation of ${\mathcal P}_1$ . The Fundamental Theorem of Projective Geometry states that any collineation of $\mathrm{PG}(hk-1, q^h)$ can be described as $(x_1,x_2, \dots, x_{hk}) \mapsto (x_1^\sigma,x_2^\sigma, \dots, x_{hk}^\sigma) A$ , where $\sigma$ is an automorphism of ${\mathbb F}_{q^h}$ and A is a non-singular matrix of order hk over ${\mathbb F}_{q^h}$ . A collineation with $\sigma = id$ is called projectivity, otherwise it is said semilinear collineation. A q-order subgeometry of $\mathrm{PG}(hk-1,q^h)$ is a pointset of $\mathrm{PG}(hk-1, q^h)$ isomorphic to the canonical $\mathrm{PG}(hk-1, q)$ , i.e., it is obtained by restricting the coordinates to ${\mathbb F}_q$ . Let $\Psi$ be the semilinear collineation of $\mathrm{PG}(hk-1,q^h)$ induced by $(x_1,x_2, \dots, x_{hk}) \mapsto (x_1^q,x_2^q, \dots, x_{hk}^q)$ . Then $\Psi$ fixes $\mathrm{PG}(hk-1,q)$ pointwise and one can prove that a d-dimensional subspace of $\mathrm{PG}(hk-1,q^h)$ is fixed by $\Psi$ if and only if it intersects the subgeometry $\mathrm{PG}(hk-1,q)$ in a d-dimensional subspace.

An $(h-1)$ -spread of the projective space $\mathrm{PG}(hk-1,q)$ is a family $\mathcal{S}$ of mutually disjoint $(h-1)$ -dimensional projective spaces such that every point of $\mathrm{PG}(hk-1,q)$ lies in exactly one element of $\mathcal{S}$ . An important class of spreads, called Desarguesian spreads, has been described in [ Reference Segre39 ] and can be constructed as follows. Embed $\Sigma \simeq \mathrm{PG}(hk-1,q)$ as a q-order subgeometry of $\mathrm{PG}(hk-1,q^h)$ in such a way that $\Sigma$ is the set of fixed points

\begin{equation*}\Sigma = \mathrm{Fix}(\Psi)\end{equation*}

of a semilinear collineation $\Psi$ of $\mathrm{PG}(hk-1,q^h)$ of order h. For each point $P \in \mathrm{PG}(hk-1,q^h)$ , define

(2·6) \begin{equation} X^*(P) \,:\!=\, \big\langle P, P^{\Psi}, \ldots, P^{\Psi^{h-1}} \big\rangle_{q^h}.\end{equation}

Note that for any point $P \in \mathrm{PG}(hk-1,q^h)$ , the subspace $X^*(P)$ has dimension at most $h-1$ .

Let $\Theta$ be a $(k-1)$ -dimensional subspace of $\mathrm{PG}(hk-1,q^h)$ such that

\begin{equation*} \big\langle \Theta, \Theta^{\Psi}, \ldots, \Theta^{\Psi^{h-1}} \big\rangle_{q^h} = \mathrm{PG}(hk-1,q^h).\end{equation*}

It follows that the space $X^*(P)$ is an $(h-1)$ -dimensional projective space of $\mathrm{PG}(hk-1,q^h)$ , for every point $P \in \Theta$ ; in other words, P is an imaginary point of $\mathrm{PG}(hk-1,q^h)$ .

By construction, for every $P \in \Theta$ , the subspace $X^*(P)$ is fixed by $\Psi$ , and hence, by Lemma 2·4,

\begin{equation*}X(P) \,:\!=\, X^*(P) \cap \Sigma\end{equation*}

is an $(h-1)$ -dimensional projective space of $\Sigma$ . As P varies in $\Theta$ , we obtain the set

\begin{equation*}\mathcal{D}(\Theta) \,:\!=\, \{ X(P) \,:\, P \in \Theta \},\end{equation*}

which consists of $q^{h(k-1)} + q^{h(k-2)} + \cdots + q^h + 1$ mutually disjoint $(h-1)$ -dimensional projective spaces of $\Sigma$ , and hence forms an $(h-1)$ -spread of $\Sigma$ . Two $(h-1)$ -spreads ${\mathcal S}_1$ and ${\mathcal S}_2$ of $\mathrm{PG}(hk-1,q)$ are equivalent if there exists a collineation of $\mathrm{PG}(hk-1, q)$ mapping one to the other. Any $(h-1)$ -spread of the projective space $\mathrm{PG}(hk-1,q)$ that is equivalent to the construction $\mathcal{D}(\Theta)$ is called a Desarguesian spread of $\mathrm{PG}(hk-1,q)$ . The $(h-1)$ -dimensional projective spaces $\Theta, \Theta^{\Psi}, \ldots, \Theta^{\Psi^{h-1}}$ are uniquely determined by the spread $\mathcal{D}(\Theta)$ . More precisely, if $\mathcal{D}(\Theta) = \mathcal{D}(\Theta')$ for some $(k-1)$ -dimensional subspace $\Theta'$ of $\mathrm{PG}(hk-1,q^h)$ , then necessarily $\Theta' = \Theta^{\Psi^i}$ for some $i \in \{0,\ldots,h-1\}$ . For this reason, the subspaces $\Theta, \Theta^{\Psi}, \ldots, \Theta^{\Psi^{h-1}}$ are referred to as the director spaces of $\mathcal{D}(\Theta)$ . For further details, see [ Reference Bader and Lunardon2 , Reference Lunardon31 , Reference Segre39 ].

Let $\Sigma'$ be the subgeometry of $\mathrm{PG}(hk-1, q^h)$ isomorphic to $\mathrm{PG}(hk-1, q)$ consisting of the points represented by

\begin{align*} & \Phi({\boldsymbol x}) = \left(x_1, x_1^q, \dots, x_1^{q^{h-1}}, x_2,x_2^q, \dots, x_2^{q^{h-1}}, \dots, x_k, x_k^q, \dots, x_k^{q^{h-1}}\right)\!,\end{align*}

where ${\boldsymbol x} = (x_1, \dots, x_k) \in {\mathbb F}_{q^h}^k$ , with $(x_1, \dots, x_k) \ne (0, \dots, 0)$ . Let $\Theta$ be the $(k-1)$ -dimensional subspace of $\mathrm{PG}(hk-1,q^h)$ given by

\begin{align*} X_{2+i} = \ldots = X_{h+i} = 0, \qquad i = 0,h,2h, \dots, h(k-1).\end{align*}

In this case, $\Sigma'$ is the set of fixed points of the semilinear collineation $\Psi'$ of $\mathrm{PG}(hk-1,q^h)$ of order h, defined by

\begin{equation*}\Psi'\,:\,\langle (y_1,\ldots,y_h,y_{h+1},\ldots,y_{2h},\ldots,y_{h(k-1)+1},\ldots,y_{hk}) \rangle_{{\mathbb F}_{q^h}}\longmapsto \end{equation*}

\begin{equation*}\big\langle \big(y_{h}^q,y_1^q,\ldots,y_{h-1}^q,y_{2h}^q,y_{h+1}^q,\ldots,y_{2h-1}^q,\ldots,y_{hk}^q,y_{h(k-1)+1}^q\ldots,y_{hk-1}^q\big) \big\rangle_{{\mathbb F}_{q^h}}.\end{equation*}

2·2·1 Automorphism group

Here we briefly recall the subgroup of $\mathrm{PGL}(hk, q)$ leaving invariant the $(h-1)$ -Desarguesian spread ${\mathcal D}(\Theta)$ . The projectivities induced by the matrix

(2·7) \begin{align}\begin{pmatrix} A_{11}\;\;\;\;\; & A_{12}\;\;\;\;\; & \dots\;\;\;\;\; & A_{1k} \\[5pt] A_{21}\;\;\;\;\; & A_{22}\;\;\;\;\; & \dots\;\;\;\;\; & A_{2k} \\[5pt] \vdots\;\;\;\;\; & \vdots\;\;\;\;\; & \ddots\;\;\;\;\; & \vdots \\[5pt] A_{k1}\;\;\;\;\; & A_{k2}\;\;\;\;\; & \dots\;\;\;\;\; & A_{kk}\end{pmatrix}\!, \end{align}

where

\begin{align*}A_{ij} =\begin{pmatrix} a_{ij}\;\;\;\;\; & 0\;\;\;\;\; & \dots\;\;\;\;\; & 0 \\[5pt] 0\;\;\;\;\; & a_{ij}^q\;\;\;\;\; & \dots\;\;\;\;\; & 0 \\[5pt] \vdots\;\;\;\;\; & \vdots\;\;\;\;\; &\ddots\;\;\;\;\; & \vdots \\[5pt] 0\;\;\;\;\; & 0\;\;\;\;\; & \dots\;\;\;\;\; & a_{ij}^{q^{h-1}}\end{pmatrix}\!,\end{align*}

for some $A = (a_{ij}) \in \mathrm{GL}(k, q^h)$ , fix $\Sigma'$ , ${\mathcal D}(\Theta)$ and each of its director spaces. They give rise to a group of projectivities ${\mathcal G}_0$ with structure

\begin{align*} \frac{{\mathrm{GL}}(k, q^h)}{Z({\mathrm{GL}}(k, q))}.\end{align*}

Lemma 2·6. The stabiliser in ${\mathcal G}_0$ of a point P of $\Sigma'$ fixes pointwise the member of ${\mathcal D}(\Theta)$ containing P.

Proof. Let g be a projectivity of ${\mathcal H}$ induced by (2·7) and fixing the point P of $\Sigma'$ represented by $\Phi({\boldsymbol x})$ . Since the image of P under g is represented by $\Phi(A{\boldsymbol x})$ and P is fixed by g, it follows that $\Phi(A{\boldsymbol x})$ and $\Phi({\boldsymbol x})$ represent the same point of $\Sigma'$ . Hence $A{\boldsymbol x} = \rho {\boldsymbol x}$ , for some $\rho \in {\mathbb F}_{q^h}^*$ . Let i, with $1 \le i \le k$ , be such that the i-th entry of ${\boldsymbol x}$ is non-zero, i.e., $x_i \ne 0$ , and denote by $(A{\boldsymbol x})_i$ the i-th entry of $A{\boldsymbol x}$ . Then on the one hand $(A{\boldsymbol x})_i^q = \rho {\boldsymbol x}_i^q$ , since $\Phi(A{\boldsymbol x})$ and $\Phi({\boldsymbol x})$ represent the same point of $\Sigma'$ . On the other hand, $(A{\boldsymbol x})_i^q = \rho^q {\boldsymbol x}_i^q$ , due to the fact that $A{\boldsymbol x} = \rho {\boldsymbol x}$ . Therefore $\rho^q = \rho$ , i.e., $\rho \in {\mathbb F}_q^*$ . A point Q of $\Sigma'$ belonging to the member of ${\mathcal D}(\Theta)$ containing P is represented by $\Phi(\lambda {\boldsymbol x})$ , for some $\lambda \in {\mathbb F}_{q^h}^*$ . Hence the image of Q under g is represented by $\Phi(A(\lambda {\boldsymbol x}))$ , where $A(\lambda {\boldsymbol x}) = \lambda A {\boldsymbol x} = \lambda \rho {\boldsymbol x}$ . Moreover, since $\rho \in {\mathbb F}_q^*$ , $\Phi(\lambda {\boldsymbol x})$ and $\Phi(\lambda \rho {\boldsymbol x})$ represent the same point of $\Sigma'$ . Consequently, Q is fixed by g, as required.

The projectivity induced by

\begin{align*}\begin{pmatrix} B\;\;\;\;\; & \mathbf{0}\;\;\;\;\; & \dots\;\;\;\;\; & \mathbf{0} \\[5pt] \mathbf{0}\;\;\;\;\; & B\;\;\;\;\; & \dots\;\;\;\;\; & \mathbf{0} \\[5pt] \vdots\;\;\;\;\; & \vdots\;\;\;\;\; & \ddots\;\;\;\;\; & \vdots \\[5pt] \mathbf{0}\;\;\;\;\; & \mathbf{0}\;\;\;\;\; & \dots\;\;\;\;\; & B\end{pmatrix}\!,\end{align*}

where

\begin{align*}B =\begin{pmatrix} \mathbf{0}\;\;\;\;\; & I_{h-1} \\[2pt] 1\;\;\;\;\; & \mathbf{0}\end{pmatrix}\end{align*}

generates a cyclic group $C_h$ of order h fixing both $\Sigma'$ , ${\mathcal D}(\Theta)$ and permuting the director spaces of ${\mathcal D}(\Theta)$ in a single orbit. They give rise to a group of projectivities ${\mathcal G}$ with structure

\begin{align*} \frac{{\mathrm{GL}}(k, q^h)}{Z({\mathrm{GL}}(k, q))} \rtimes C_h.\end{align*}

Actually, it turns out that ${\mathcal G}$ contains all the projectivities of $\Sigma'$ leaving ${\mathcal D}(\Theta)$ invariant, see [ Reference Dye16 , Reference Van de Voorde50 ]. In particular, the orbit of a director space of ${\mathcal D}(\Theta)$ under ${\mathcal G}$ has size h.

2·2·2 Quadrics

Let ${\mathbb F}$ be a finite field and let V be an ${\mathbb F}$ -vector space. A quadratic form over ${\mathbb F}$ is a map $\mathcal{Q}\colon V\to{\mathbb F}$ such that:

1. (1) $\mathcal{Q}(\alpha v)=\alpha^2\mathcal{Q}(v)$ , for all $\alpha\in{\mathbb F},\ v\in V$ ;

2. (2) the map $B(u,v)=\mathcal{Q}(u+v)-\mathcal{Q}(u)-\mathcal{Q}(v)$ is bilinear.

The set of points $P=\langle v \rangle_{{\mathbb F}}$ of the projective space $\mathrm{PG}(V,{\mathbb F})$ that satisfy $\mathcal{Q}(v)=0$ is called quadric (or quadratic variety) of $\mathrm{PG}(V, {\mathbb F})$ associated with $\mathcal{Q}$ . By abuse of notation, we denote both the quadratic form and the corresponding quadric by $\mathcal{Q}$ .

Let $\Sigma'$ be the subgeometry of $\mathrm{PG}(hk-1, q^h)$ isomorphic to $\mathrm{PG}(hk-1, q)$ consisting of the points represented by

\begin{align*} & \left(x_1, x_1^q, \dots, x_1^{q^{h-1}}, x_2,x_2^q, \dots, x_2^{q^{h-1}}, \dots, x_k, x_k^q, \dots, x_k^{q^{h-1}}\right)\!,\end{align*}

where $x_i \in {\mathbb F}_{q^h}$ , for $i = 1, \dots, k$ , with $(x_1, \dots, x_k) \ne (0, \dots, 0)$ . Let $\Theta$ be the $(k-1)$ -dimensional subspace of $\mathrm{PG}(hk-1,q^h)$ given by

\begin{align*} X_{2+i} = \ldots = X_{h+i} = 0, \qquad i = 0,h,2h, \dots, h(k-1).\end{align*}

Starting from a quadric $\mathcal{Q}$ of $\Theta$ , it is possible to construct a family of quadrics of $\Sigma'$ by means of the so–called trace trick. More precisely, let $\mathrm{Tr}_{q^h/q}$ denote the trace map from ${\mathbb F}_{q^h}$ to ${\mathbb F}_q$ and, for $\alpha\in{\mathbb F}_{q^h}$ , define

\begin{equation*}L_\alpha\colon {\mathbb F}_{q^h} \to {\mathbb F}_{q}, \qquad x\mapsto \mathrm{Tr}_{q^h/q}(\alpha x).\end{equation*}

Then, it is easy to check that the composition $L_\alpha\circ\mathcal{Q}\colon V\to{\mathbb F}$ is a quadratic form over ${\mathbb F}_q$ . Therefore, to any quadric $\mathcal{Q}$ of $\Theta$ one can naturally associate a system of quadrics $L_\alpha\circ\mathcal{Q}, \alpha\in{\mathbb F}_{q^h}$ of $\Sigma'$ . For further details on this construction, see for instance [ Reference Gill18 , Reference Lavrauw and Van de Voorde29 ].

A set of projective subspaces is said to be an intersection of quadrics if there exists a family of quadrics whose intersection contains each of its subspaces. It is called a complete intersection of quadrics if that intersection contains no further points. Here, we note that if a point set ${\mathcal Z} \subset \Theta \simeq \mathrm{PG}(k-1, q^h)$ is intersection of a family of quadrics, then the members of

\begin{equation*}X({\mathcal Z}) = \{X(P) \;:\; P \in {\mathcal Z}\}\end{equation*}

are still contained in the intersection of a suitable family of quadrics. Moreover, $X({\mathcal Z})$ is a complete intersection of quadrics if ${\mathcal Z}$ does.

Proposition 2·7. Let ${\mathcal Z} \subseteq \Theta \simeq \mathrm{PG}(k-1,q^h)$ be an intersection of quadrics of $\Theta$ . Then $X(\mathcal{Z})$ is an intersection of quadrics of $\Sigma' \simeq \mathrm{PG}(hk-1,q)$ . Moreover, if $\mathcal{Z}$ is a complete intersection of quadrics, then $X(\mathcal{Z})$ is a complete intersection of quadrics as well.

Proof. Let $\mathcal{Q}_1,\ldots,\mathcal{Q}_{\ell}$ be quadrics of $\Theta \simeq \mathrm{PG}(k-1,q^h)$ such that $\mathcal{Z} \subseteq \bigcap_{i=1}^{\ell} \mathcal{Q}_i$ . This means that for every point $P=\langle v \rangle_{{\mathbb F}_{q^h}} \in \mathcal{Z}$ , we have $\mathcal{Q}_i(v)=0$ , for all $i=1,\ldots,\ell$ . For each $\alpha \in {\mathbb F}_{q^h}^*$ , consider the quadratic form $L_\alpha \circ \mathcal{Q}_i$ obtained via the trace $\mathrm{Tr}_{q^h/q}$ , which defines a quadric of $\Sigma' \simeq \mathrm{PG}(hk-1,q)$ . Since $\mathcal{Q}_i(v)=0$ , we also have

\begin{equation*}(L_\alpha \circ \mathcal{Q}_i)(v) = 0\quad \text{for all } \alpha \in {\mathbb F}_{q^h}^*.\end{equation*}

Therefore, the $(h-1)$ -dimensional projective space X(P) associated with P is contained in every quadric $L_\alpha \circ \mathcal{Q}_i$ , and hence

(2·8) \begin{equation} X(P) \subseteq \bigcap_{i=1}^{\ell} \bigcap_{\alpha \in {\mathbb F}_{q^h}^*} (L_\alpha \circ \mathcal{Q}_i).\end{equation}

Since this holds for every $P \in \mathcal{Z}$ , it follows that $X(\mathcal{Z})$ is an intersection of quadrics of $\Sigma' \simeq \mathrm{PG}(hk-1,q)$ .

Assume now that $\mathcal{Z}$ is a complete intersection of quadrics, that is, $\mathcal{Z} = \bigcap_{i=1}^{\ell} \mathcal{Q}_i$ . By the first part of the proof, we have that (2·8) holds, for every $P \in \mathcal{Z}$ . Let $R=\langle w\rangle_{{\mathbb F}_q}$ be a point of $\Sigma' \simeq \mathrm{PG}(hk-1,q)$ belonging to the right-hand side of (2·8). Then R lies on every quadric $L_{\alpha}\circ \mathcal{Q}_i$ , for all $\alpha\in{\mathbb F}_{q^h}^*$ and for all i. By definition of $L_\alpha$ , this yields

\begin{equation*}\mathrm{Tr}_{{q^h}/q}(\alpha\,\mathcal{Q}_i(w))=0,\quad \text{for all } \alpha\in{\mathbb F}_{q^h}^*.\end{equation*}

Since the trace form is non-degenerate, it follows that $\mathcal{Q}_i(w)=0,$ and this holds for every $i=1,\ldots,\ell$ . Hence the point $R^*=\langle w \rangle_{{\mathbb F}_{q^h}} \in \Theta \simeq \mathrm{PG}(k-1,q^h)$ belongs to $\bigcap_{i=1}^{\ell} \mathcal{Q}_i=\mathcal{Z}$ . Consequently, $R \in X(R^*)$ , with $R^* \in \mathcal{Z}$ , proving that $X(\mathcal{Z})$ is complete intersection of the quadrics $L_{\alpha} \circ \mathcal{Q}_i$ , $\alpha \in {\mathbb F}_{q^h}^*$ , $i=1,\ldots,\ell$ , in $\Sigma' \simeq \mathrm{PG}(hk-1,q)$ .

3·1. Construction

Consider a normal rational curve $\mathcal{N}_{hk,q^h}$ in $\mathrm{PG}(hk-1,q^h)$ . For an imaginary point P of $\mathcal{N}_{hk,q^h}$ , the subspace $X^*(P) = \langle P, P^{\Psi}, \ldots, P^{\Psi^{h-1}} \rangle_{{\mathbb F}_{q^h}}$ is an $(h-1)$ -dimensional projective subspace of $\mathrm{PG}(hk-1,q^h)$ . We now consider the projective subspaces X(P) obtained by intersecting $X^*(P)$ with the subgeometry $\Sigma$ . We show that these subspaces form a pseudo-arc in $\mathrm{PG}(hk-1,q)$ .

Proposition 3·1. Let $P_1, \ldots, P_{\ell}$ be imaginary points of $\mathcal{N}_{hk,q^h}$ in $\mathrm{PG}(hk-1,q^h)$ , no two of which lie in the same orbit under the cyclic group generated by $\Psi$ . Assume that $\Psi$ stabilises $\mathcal{N}_{hk,q^h}$ and that $\ell \geq k$ . Then the set

\begin{equation*}\{X(P_i), \ i = 1, \ldots, \ell\}\end{equation*}

is a pseudo-arc in $\Sigma$ .

Proof. Since the points $P_1, \ldots, P_{\ell}$ are imaginary points of $\mathcal{N}_{hk,q^h}$ , it follows from the definition and from Lemma 2·4 that each $X(P_i)$ is an $(h-1)$ -dimensional projective space of $\Sigma$ . Suppose, by contradiction, that k of them do not span the whole space $\Sigma$ . Then, without loss of generality, we may assume that $X(P_1), \ldots, X(P_k)$ are contained in a hyperplane of $\Sigma$ . This would imply that the points

\begin{equation*}P_1, P_1^{\Psi}, \ldots, P_1^{\Psi^{h-1}},P_2, P_2^{\Psi}, \ldots, P_2^{\Psi^{h-1}},\ldots,P_k, P_k^{\Psi}, \ldots, P_k^{\Psi^{h-1}}\end{equation*}

are contained in a hyperplane of $\mathrm{PG}(hk-1,q^h)$ . However, since $P_1, \ldots, P_k$ belong to distinct orbits under the cyclic group generated by $\Psi$ , all the above points are distinct. Thus we obtain hk distinct points of the normal rational curve $\mathcal{N}_{hk,q^h}$ , that are contained in a hyperplane of $\mathrm{PG}(hk-1,q^h)$ , contradicting the fact that ${\mathcal N}_{hk, q^h}$ is an arc of $\mathrm{PG}(hk-1,q^h)$ . Therefore, every k of the subspaces $X(P_i)$ span $\Sigma$ , and the claim follows.

From the above result, the question of determining normal rational curves $\mathcal{N}_{hk,q^h}$ stabilised by $\Psi$ naturally arises. We consider it in the following.

Proof. Recall that $\Sigma=\mathrm{Fix}(\Psi)$ , that is,

(3·1) \begin{equation}P\in \Sigma \iff P^\Psi=P.\end{equation}

Assume first that ${\mathcal N}_{hk,q^h}\cap\Sigma={\mathcal N}_{hk,q}$ . Then ${\mathcal N}_{hk,q}\subseteq {\mathcal N}_{hk,q^h}$ and ${\mathcal N}_{hk,q}$ is pointwise fixed by $\Psi$ , by (3·1). Moreover, since $\lvert {\mathcal N}_{hk,q} \rvert = q+1 \ge hk+2$ , by [ Reference Hirschfeld and Thas26 , theorem 6·30], ${\mathcal N}_{hk,q^h}$ is the unique normal rational curve of $\mathrm{PG}(hk-1,q^h)$ containing ${\mathcal N}_{hk,q}$ . Since $\Psi$ is a collineation of $\mathrm{PG}(hk-1,q^h)$ , it follows that ${\mathcal N}_{hk,q^h}^{\Psi}$ is also a normal rational curve containing ${\mathcal N}_{hk,q}$ ; hence ${\mathcal N}_{hk,q^h}^{\Psi}={\mathcal N}_{hk,q^h}$ .

Conversely, assume that ${\mathcal N}_{hk,q^h}^{\Psi}={\mathcal N}_{hk,q^h}$ . Let $\nu_{hk,q^h}\,:\,\mathrm{PG}(1,q^h)\to \mathrm{PG}(hk-1,q^h)$ be the Veronese embedding and set

\begin{equation*}{\mathcal N}'_{hk,q^h}\,:\!=\,\{\nu_{hk,q^h}(u,t)\,:\, \langle(u,t)\rangle_{{\mathbb F}_{q^h}}\in\mathrm{PG}(1,q^h)\}\subseteq \mathrm{PG}(hk-1,q^h).\end{equation*}

Every normal rational curve of $\mathrm{PG}(hk-1,q^h)$ is projectively equivalent to ${\mathcal N}'_{hk,q^h}$ . Hence there exists $A\in {\mathrm{GL}}(hk,q^h)$ such that

\begin{equation*}{\mathcal N}_{hk,q^h}=\big\{A\, \nu_{hk,q^h}(u,t)^{\top}\,:\, \langle(u,t)\rangle_{{\mathbb F}_{q^h}}\in\mathrm{PG}(1,q^h)\big\}.\end{equation*}

Since ${\mathcal N}_{hk,q^h}$ is $\Psi$ –invariant, for every $\langle(u,t)\rangle\in\mathrm{PG}(1,q^h)$ there exists a (necessarily unique) point $\langle(u',t')\rangle\in\mathrm{PG}(1,q^h)$ such that

(3·2) \begin{equation}A^{q}\, \nu_{hk,q^h}(u^q,t^q)^{\top}=\big(A\, \nu_{hk,q^h}(u,t)^{\top}\big)^{\Psi}=A\, \nu_{hk,q^h}(u',t')^{\top}.\end{equation}

Multiplying (3·2) on the left by $A^{-1}$ yields

\begin{equation*}S\, \nu_{hk,q^h}(u^q,t^q)^{\top}=\nu_{hk,q^h}(u',t')^{\top},\quad \text{where} \quad S\,:\!=\,A^{-1}A^{q}\in {\mathrm{GL}}(hk,q^h).\end{equation*}

Thus the projectivity induced by S stabilises the standard curve ${\mathcal N}'_{hk,q^h}$ . By [ Reference Hirschfeld and Thas26 , theorem 6·32], there exists a matrix $B\in {\mathrm{GL}}(2,q^h)$ such that the action of S on ${\mathcal N}'_{hk,q^h}$ is induced by B, that is, for every $\langle(u,t)\rangle_{{\mathbb F}_{q^h}}\in\mathrm{PG}(1,q^h)$ ,

(3·3) \begin{equation}\nu_{hk,q^h}\big(B\, (u,t)^{\top}\big)=S\, \nu_{hk,q^h}(u,t)^{\top}.\end{equation}

We now determine ${\mathcal N}_{hk,q^h}\cap \Sigma$ . Let $P=A\, \nu_{hk,q^h}(u,t)^{\top}\in {\mathcal N}_{hk,q^h}$ . By (3·1), $P\in\Sigma$ if and only if $P^\Psi=P$ , that is,

(3·4) \begin{equation}S\, \nu_{hk,q^h}(u^q,t^q)^{\top}=\nu_{hk,q^h}(u,t)^{\top}.\end{equation}

Note that, by definition of S, we have the identity

\begin{equation*}S\,S^{q}\cdots S^{q^{h-1}}=I.\end{equation*}

Equivalently, by (3·3), the matrix $B\in {\mathrm{GL}}(2,q^h)$ (defined up to scalar multiplication) satisfies

\begin{equation*}B\,B^{q}\cdots B^{q^{h-1}}=I.\end{equation*}

By Hilbert’s Theorem 90 (see, for example, [ Reference Serre42 , proposition 3, p. 151] and [ Reference Glasby and Howlett19 , proposition 1·3]), there exists $C\in {\mathrm{GL}}(2,q^h)$ such that

(3·5) \begin{equation}B=C^{-1}C^{q}.\end{equation}

Substituting (3·5) into (3·3) and using (3·4), we obtain

\begin{equation*}\nu_{hk,q^h}\big(C^{-1}C^{q}(u^q,t^q)^{\top}\big)=\nu_{hk,q^h}(u,t),\end{equation*}

which implies that $\langle(u,t)\rangle\in \mathrm{PG}(1,q)$ .

Therefore, ${\mathcal N}_{hk,q^h}\cap\Sigma$ is the image under $A \, \nu_{hk,q^h}$ of a $\mathrm{PG}(1,q)$ –subline. Consequently, ${\mathcal N}_{hk,q^h}\cap\Sigma$ is a normal rational curve in $\Sigma$ .

At this stage, we need to compute the number of imaginary points of a normal rational curve of $\mathrm{PG}(hk-1,q^h)$ fixed by $\Psi$ . Taking into account Proposition 3·2, we can represent the normal rational curve as in (2·3):

\begin{equation*}\mathcal{N}_{hk,q^h}=\bigl\{\langle (1, t, t^2, \ldots, t^{hk-1}) \rangle_{{\mathbb F}_{q^h}} \,:\, t \in {\mathbb F}_{q^h}\bigr\}\cup\bigl\{\langle (0, \ldots, 0, 1) \rangle_{{\mathbb F}_{q^h}}\bigr\}.\end{equation*}

We next determine the conditions under which a point of the normal rational curve is imaginary.

Lemma 3·3. The imaginary points of $\mathcal{N}_{hk,q^h}$ are precisely the points of the form

(3·6) \begin{equation} P_{\alpha} = \langle (1, \alpha, \alpha^2, \ldots, \alpha^{hk-1}) \rangle_{{\mathbb F}_{q^h}},\end{equation}

where $\alpha \in {\mathbb F}_{q^h}$ satisfies ${\mathbb F}_q(\alpha) = {\mathbb F}_{q^h}$ .

Proof. For any $\alpha \in {\mathbb F}_{q^h}$ and $i=0,\ldots,h-1$ , we have

\begin{equation*}P_{\alpha}^{\Psi^i}=\big\langle \big(1, \alpha^{q^i}, \alpha^{2q^i}, \ldots, \alpha^{(hk-1)q^i}\big) \big\rangle_{{\mathbb F}_{q^h}}.\end{equation*}

Consider the matrix whose rows are representatives of the points $P_{\alpha}, P_{\alpha}^{\Psi}, \ldots, P_{\alpha}^{\Psi^{h-1}}$ , namely

\begin{equation*}M_\alpha=\begin{pmatrix}1\;\;\;\;\; & \alpha\;\;\;\;\; & \alpha^2\;\;\;\;\; & \cdots\;\;\;\;\; & \alpha^{hk-1}\\[3pt] 1\;\;\;\;\; & \alpha^{q}\;\;\;\;\; & \alpha^{2q}\;\;\;\;\; & \cdots\;\;\;\;\; & \alpha^{(hk-1)q}\\[3pt] \vdots\;\;\;\;\; & \vdots\;\;\;\;\; & \vdots\;\;\;\;\; & & \vdots\\[3pt] 1\;\;\;\;\; & \alpha^{q^{h-1}}\;\;\;\;\; & \alpha^{2q^{h-1}}\;\;\;\;\; & \cdots\;\;\;\;\; & \alpha^{(hk-1)q^{h-1}}\end{pmatrix} \in {\mathbb F}_{q^h}^{h\times hk}.\end{equation*}

This is a (rectangular) q-Moore matrix associated with the elements $1,\alpha,\alpha^2,\ldots,\alpha^{hk-1}$ . By [ Reference Lidl and Niederreiter30 , lemma 3·51], we get that the rows of $M_\alpha$ are linearly independent over ${\mathbb F}_{q^h}$ if and only if $\alpha$ does not belong to any proper subfield of ${\mathbb F}_{q^h}$ , that is, if and only if ${\mathbb F}_q(\alpha)={\mathbb F}_{q^h}$ . Consequently, the points

\begin{equation*}P_{\alpha}, P_{\alpha}^{\Psi}, \ldots, P_{\alpha}^{\Psi^{h-1}}\end{equation*}

span an $(h-1)$ -dimensional projective space if and only if $\alpha$ generates ${\mathbb F}_{q^h}$ over ${\mathbb F}_q$ . This proves the claim.

In view of Proposition 3·1, we consider imaginary points that do not belong to the same orbit under the cyclic group generated by $\Psi$ . It is straightforward to verify that this condition corresponds to selecting elements $\alpha \in {\mathbb F}_{q^h}$ lying in distinct orbits of the q-Frobenius automorphism. For this reason, we define $\Lambda_{h,q}$ to be a set of representatives of the orbits of size h of ${\mathbb F}_{q^h}$ under the q-Frobenius automorphism. Note that $\alpha \in \Lambda_{h,q}$ if and only if ${\mathbb F}_q(\alpha) = {\mathbb F}_{q^h}$ . The size of $\Lambda_{h,q}$ coincides with the number of monic irreducible polynomials over ${\mathbb F}_q$ of degree h and is given by

\begin{equation*} |\Lambda_{h,q}| = \frac{1}{h}\sum_{d \mid h} \mu\!\left(\frac{h}{d}\right) q^d,\end{equation*}

where $\mu$ denotes the Möbius function, defined for $m \in \mathbb{N}$ by

\begin{equation*} \mu(m)=\begin{cases} 1 & \mbox{ if } m=1, \\ ({-}1)^k & \mbox{ if } m \mbox{ is the product of } k \mbox{ distinct primes, } \\ 0 & \mbox{ if } m \mbox{ is divisible by the square of a prime.} \end{cases}\end{equation*}

See for instance [ Reference Hirschfeld25 , section 1·6]. The size of ${\Lambda _{h,q}}$ can be expressed as

\begin{equation*} |\Lambda_{h,q}| = \frac{1}{h}\sum_{d\mid h}\mu\!\left(\frac{h}{d}\right)q^d = \frac{q^h}{h} + \frac{1}{h} \sum_{\substack{d\mid h\\ d \lt h}} \mu\!\left(\frac{h}{d}\right)q^d. \end{equation*}

In particular, for fixed $h$ , the dominant term is ${q^h}/h$ , and hence

\begin{equation*} |\Lambda_{h,q}| \sim \frac{q^h}{h} \quad \text{as }q\to\infty. \end{equation*}

Thus, for fixed $h$ , the quantity $|{\Lambda _{h,q}}|$ has the same asymptotic order as ${q^h}$ . In particular, $|\Lambda_{h,q}|=\mathcal{O}(q^h)$ , as q grows. Finally, we observe that, when h is prime, the expression for $|\Lambda_{h,q}|$ simplifies to

\begin{equation*}|\Lambda_{h,q}| = \frac{q^h - q}{h}.\end{equation*}

The following result holds true.

Theorem 3·4. The set

(3·7) \begin{equation} \mathcal{P}_{h,k,q}\,:\!=\,\{X(P_{\alpha}) = X^*(P_{\alpha}) \cap \Sigma, \ \alpha \in \Lambda_{h,q}\},\end{equation}

where $P_{\alpha}$ is as in (3·6), is a pseudo-arc in $\Sigma \simeq \mathrm{PG}(hk-1,q)$ having size $|\Lambda_{h,q}|$ .

3·2. The pseudo-arc $\mathcal{P}_{h,k,q}$ is non-Desarguesian

Let ${\mathcal D}$ be a Desarguesian $(h-1)$ -spread of $\Sigma$ with director spaces $\Theta, \Theta^{\Psi}, \dots, \Theta^{\Psi^{h-1}}$ . A natural way to construct a pseudo-arc of $\Sigma$ is the following. Consider an arc ${\mathcal{A}}$ in $\Theta \simeq {\rm{PG}}(k - 1,{q^h})$ . The subset of ${\mathcal D}$ given by

\begin{equation*}X({\mathcal A}) = \{X(P) \; : \; P \in {\mathcal A}\}\end{equation*}

is a pseudo-arc of $\Sigma$ . Since all the elements of a pseudo-arc obtained from an arc ${\mathcal A}$ in this way belong to the Desarguesian $(h-1)$ -spread ${\mathcal D}$ of $\Sigma$ , we will refer to such a pseudo-arc as a Desarguesian pseudo-arc. In this section, we prove that the pseudo-arc $\mathcal{P}_{h,k,q}$ is non-Desarguesian, that is, there is no Desarguesian $(h-1)$ -spread of $\Sigma$ containing all its members.

Consider the setting as in Section 3·1. Let ${\mathcal N}_{hk, q^h}$ denote the normal rational curve of $\mathrm{PG}(hk-1, q^h)$ , $q \ge hk+1$ , as in (2·3). Then ${\mathcal N}_{hk, q^h} \cap \Sigma = {\mathcal N}_{hk, q}$ , where $\Sigma$ is the canonical subgeometry $\mathrm{PG}(hk-1, q)$ . Let H be the subgroup of $\mathrm{PGL}(hk, q)$ of order $(q^3-q)/d$ , where $d = \gcd(2, q-1)$ , isomorphic to $\mathrm{PSL}(2, q)$ acting on ${\mathcal N}_{hk, q}$ , see Lemma 2·3.

Lemma 3·6. The group H fixes the pseudo-arc

\begin{align*} {\mathcal P}_{h,k,q} = \{ X(P_{\alpha}) = X^*(P_{\alpha}) \cap \Sigma \;:\; \alpha \in \Lambda_{h,q} \}.\end{align*}

Proof. Assume, by contradiction, that ${\mathcal D}(\Theta)$ is a Desarguesian $(h-1)$ -spread fixed by H. Then H is a subgroup of the group of projectivities ${\mathcal G}$ of $\Sigma$ leaving invariant ${\mathcal D}(\Theta)$ . See Section 2·2·1, for a description of ${\mathcal G}$ . Since the orbit of a director space of ${\mathcal D}(\Theta)$ under ${\mathcal G}$ has size h, it follows that $\Theta^H$ , the orbit of $\Theta$ under the action of H, has size at most h. Hence the stabiliser in H of $\Theta$ is a subgroup of H having order $|H|/|\Theta^H|$ . Since $q \ge hk+1 \ge 2h+1$ , we have that

\begin{align*} \frac{|H|}{|\Theta^H|} \ge \frac{|H|}{h} \ge \frac{2|H|}{q-1} = \frac{2q(q+1)}{d}.\end{align*}

By [ Reference Suzuki45 , theorem 6·25], there are no proper subgroups of H of order greater than or equal to ${2q(q+1)}/{d}$ . Therefore, $\Theta$ is fixed by H. In particular, H is a subgroup of ${\mathcal G}_0$ . Next, denote by C the subgroup of H of order $q(q-1)/d$ fixing the point $P_1 = \langle (1,0, \dots, 0) \rangle_{{\mathbb F}_q}$ of ${\mathcal N}_{hk, q}$ . By Lemma 2·6, the group C has to fix pointwise the member of ${\mathcal D}(\Theta)$ containing $P_1$ , say $\Gamma$ . An element of C is induced by the matrix

\begin{align*} M(a,b) = \begin{pmatrix} a^{hk-1}\;\;\;\;\; & \binom{hk-1}{1} a^{hk-2}b\;\;\;\;\; & \binom{hk-1}{2} a^{hk-3}b^2\;\;\;\;\; & \cdots\;\;\;\;\; & b^{hk-1}\\[3pt]0\;\;\;\;\; & a^{hk-3}\;\;\;\;\; & \binom{hk-2}{1}a^{hk-4}b\;\;\;\;\; & \cdots\;\;\;\;\; & b^{hk-2}a^{-1}\\[3pt]0\;\;\;\;\; & 0\;\;\;\;\; & a^{hk-5}\;\;\;\;\; & \cdots\;\;\;\;\; & b^{hk-3}a^{-2}\\[3pt]\vdots\;\;\;\;\; & \vdots\;\;\;\;\; & \vdots\;\;\;\;\; & \ddots\;\;\;\;\; & \vdots\\[3pt]0\;\;\;\;\; & 0\;\;\;\;\; & 0\;\;\;\;\; & \cdots\;\;\;\;\; & a^{1-hk} \end{pmatrix}\!,\end{align*}

for some $a \in {\mathbb F}_q^*$ , $b \in {\mathbb F}_q$ . M(a, b) sends $(1,0, \dots, 0)$ to $a^{hk-1}(1,0, \dots, 0)$ . Let $T = \langle (x_1, x_2, \dots, x_{hk}) \rangle_{{\mathbb F}_{q}}$ be a point of $\Gamma$ distinct from $P_1$ . Observe that every projectivity of C has to fix T. Then M(a, b) sends $(x_1, x_2, \dots, x_{hk})$ to $a^{hk-1}(x_1, x_2, \dots, x_{hk})$ , for all $a \in {\mathbb F}_q^*$ , $b \in {\mathbb F}_q$ . In particular M(a, 0) sends $(x_1, x_2, \dots, x_{hk})$ to $a^{hk-1}(x_1, x_2, \dots, x_{hk})$ , for all $a \in {\mathbb F}_q^*$ . Therefore, if $x_{i+1} \ne 0$ , with $0 \le i \le hk-1$ , then $a^{hk-1-2i} x_{i+1} = a^{hk-1} x_{i+1}$ implies $a^{-2i} = 1$ , for all $a \in {\mathbb F}_q^*$ , that is $q-1 \mid 2i$ , with $0 \le i \le hk-1 \le q-2$ . Two possibilities occur, either $i = 0$ , or q is odd, with $q \le 2hk-1$ , and $i = (q-1)/2$ . In the former case T and $P_1$ coincide, a contradiction. Assume that q is odd, with $q \le 2hk-1$ , and $T = \langle (x_1, 0, \dots, 0, x_{\frac{q+1}{2}}, 0, \dots, 0) \rangle_{{\mathbb F}_{q}}$ , where $x_{\frac{q+1}{2}} \ne 0$ . Note that M(1, b) sends $(1,0,\dots, 0)$ to $(1,0,\dots,0)$ . Hence, necessarily, M(1,b) sends $(x_1, 0, \dots, 0, x_{\frac{q+1}{2}}, 0, \dots, 0)$ to $(x_1, 0, \dots, 0, x_{\frac{q+1}{2}}, 0, \dots, 0)$ , for all $b \in {\mathbb F}_q$ . Hence the following hold:

(3·8) \begin{align} \binom{hk-1-i}{\frac{q-1}{2}-i} = 0, \qquad 0 \le i \le \frac{q-3}{2}. \end{align}

Set $r = hk-1-({q-1})/{2}$ . Since $({q-1})/{2} \le hk-1 \le q-2 = ({q-1})/{2} + ({q-3})/{2}$ , then $0 \le r \le ({q-3})/{2}$ , and (3·8) are equivalent to

\begin{align*} \binom{r+j}{j} = 0, \qquad 1 \le j \le \frac{q-1}{2}. \end{align*}

Let $q = p^s$ and $m = p^{s-1}$ . We claim that $\binom{r+m}{m} \ne 0$ in ${\mathbb F}_q$ . Since $p \ge 3$ and $m \ge 1$ , then $({pm-1})/{2} - m = ({(p-2)m-1})/{2} \ge 0$ and hence $({pm-1})/{2} = ({p^s-1})/{2} =({q-1})/$ ${2} \ge m$ . For a fixed r, with $0 \le r \le ({q-3})/{2}$ , let b, c integers such that $r = bm+c$ , with $0 \le c \le m-1$ . Note that $1 \le b+1 \le ({p+1})/{2}$ , otherwise, if $b \ge ({p+1})/{2}$ , then $r = bm+c \ge bm \ge ({p+1})/{2} m \gt ({q-3})/{2}$ , a contradiction. Consider the polynomial $(1+X)^{r+m}$ over ${\mathbb F}_q$ . Since $r+m = (b+1) m +c$ , we have that $(1+X)^{r+m} = (1+X)^{m(b+1)} (1+X)^c = (1+X^m)^{b+1}(1+X)^c$ . On the one hand, the coefficient of $X^m$ in $(1+X)^{r+m}$ equals $\binom{r+m}{m}$ , whereas on the other hand, since $c \lt m$ , the coefficient of $X^m$ in $(1+X^m)^{b+1}(1+X)^c$ equals $b+1$ . Therefore

\begin{align*} \binom{r+m}{m} \equiv b+1 \pmod{p}\end{align*}

and hence $\binom{r+m}{m} \ne 0$ in ${\mathbb F}_q$ . As a consequence, the group C does not stabilise T, a contradiction.

The next result has been obtained in [ Reference Cossidente, Marino, Pavese, Santonastaso and Sheekey12 ].

Combining the previous result, together with [ Reference Hirschfeld23 , p. 66], the following holds.

We are now ready to prove that the pseudo-arc $\mathcal{P}_{h,k,q}$ is non-Desarguesian.

Theorem 3·10. The pseudo-arc

\begin{align*} {\mathcal P}_{h,k,q} = \left\{ X(P_{\alpha}) = X^*(P_{\alpha}) \cap \Sigma \;:\; \alpha \in \Lambda_{h,q} \right\}\end{align*}

is non-Desarguesian.

Proof. Assume, by contradiction, that ${\mathcal D}$ is a Desarguesian $(h-1)$ -spread of $\Sigma$ such that ${\mathcal P}_{h,k,q} \subset {\mathcal D}$ . Since H fixes ${\mathcal P}_{h,k,q}$ , but there is no Desarguesian $(h-1)$ -spread left invariant by H, it follows that there is a Desarguesian $(h-1)$ -spread of $\Sigma$ , say ${\mathcal D}'$ , contained in ${\mathcal D}^H$ , with ${\mathcal D} \ne {\mathcal D}'$ , and such that ${\mathcal P}_{h,k,q} \subset {\mathcal D} \cap {\mathcal D}'$ . Therefore, by Corollary 3·9, we have

\begin{equation*}|{\mathcal P}_{h,k,q}|\le q^r+k\end{equation*}

for some proper divisor $r$ of $h$ . On the other hand,

\begin{equation*}|{\mathcal P}_{h,k,q}|=|\Lambda_{h,q}|\ge\frac{q^h-q^{h-1}}{h}=\frac{q-1}{h}q^{h-1}.\end{equation*}

Since $q \ge hk + 1$ , we have $(q - 1)/h \ge k$ , and hence

\begin{equation*}|\Lambda_{h,q}|\ge kq^{h-1}.\end{equation*}

Since $r \lt h$ , we have ${q^r} \le {q^{h - 1}}$ . Therefore

\begin{equation*}q^r+k\le q^{h-1}+k \lt kq^{h-1}\le |\Lambda_{h,q}|,\end{equation*}

where the strict inequality follows from $k \ge 2$ and ${q^{h - 1}} \gt k$ under our assumptions. This contradiction proves the assertion.

3·3. Extending the pseudo-arc $\mathcal{P}_{h,k,q}$

A pseudo-arc is said to be complete if it is not contained in a larger pseudo-arc. The study of the completeness of the normal rational curve has been addressed in a series of papers [ Reference Seroussi and Roth41 , Reference Storme43 , Reference Storme and Thas44 ]. In this section, we show that it is possible to enlarge the pseudo-arc ${\mathcal P}_{h,k,q}$ in $\Sigma$ , which arises from the normal rational curve ${\mathcal N}_{hk, q^h}$ of $\mathrm{PG}(hk-1,q^h)$ , by adding the osculating spaces of the normal rational curve ${\mathcal N}_{hk, q}$ of $\Sigma$ . Throughout, we keep the notation of Section 3·1 and assume that $\mathrm{char}({\mathbb F}_q)=p\geq h$ . Again, let ${\mathcal N}_{hk, q^h}$ denote the normal rational curve of $\mathrm{PG}(hk-1, q^h)$ , $q \ge hk-1$ , as in (2·3). Then ${\mathcal N}_{hk, q^h} \cap \Sigma = {\mathcal N}_{hk, q}$ , where $\Sigma$ is the canonical subgeometry $\mathrm{PG}(hk-1, q)$ . For each point $Q\in\mathcal{N}_{hk,q}$ , let $\mathrm{Osc}_{h-1}(Q)$ be the osculating space of order $h-1$ at Q, which is an $(h-1)$ -dimensional projective space of $\Sigma$ since $p\geq h$ . Consider the family

\begin{equation*}\mathcal{O}_{h-1}\;\,:\!=\,\{\mathrm{Osc}_{h-1}(\nu_{hk,q}(1,t)) \,:\, t\in{\mathbb F}_q\}\ \cup\ \{\mathrm{Osc}_{h-1}(\nu_{hk,q}(0,1))\},\end{equation*}

which consists of $q+1$ projective subspaces of dimension $h-1$ in $\Sigma$ .

Theorem 3·11. Assume that $\mathrm{char}({\mathbb F}_q)=p\geq h$ . Then

\begin{equation*}{\mathcal P}_{h,k,q} \cup\ \mathcal{O}_{h-1}\end{equation*}

is a pseudo-arc in $\Sigma \simeq \mathrm{PG}(hk-1,q)$ of size $|\Lambda_{h,q}|+q+1$ .

Proof. By Theorem 3·4, we know that ${\mathcal P}_{h,k,q}$ is a pseudo-arc. We need to show that adding the osculating spaces in $\mathcal{O}_{h-1}$ preserves the pseudo-arc property. For each $t\in{\mathbb F}_q$ , let $A_t$ be the matrix whose rows form a basis for the osculating space of order $h-1$ of $\mathcal{N}_{hk,q}$ at the point $\nu_{hk,q}(1,t)$ . According to (2·4), we can write

(3·9) \begin{equation} A_t \,:\!=\,\begin{pmatrix}1\;\;\;\;\; & t\;\;\;\;\; & t^{2}\;\;\;\;\; & \cdots\;\;\;\;\; & t^{hk-1} \\[4pt]0\;\;\;\;\; & 1\;\;\;\;\; & 2t\;\;\;\;\; & \cdots\;\;\;\;\; & (hk-1)t^{hk-2} \\[4pt]0\;\;\;\;\; & 0\;\;\;\;\; & 2\!\cdot\!1\;\;\;\;\; & \cdots\;\;\;\;\; & (hk-1)(hk-2)t^{hk-3} \\[4pt]\vdots\;\;\;\;\; & \vdots\;\;\;\;\; & \vdots\;\;\;\;\; & \ddots\;\;\;\;\; & \vdots \\[4pt]0\;\;\;\;\; & \cdots\;\;\;\;\; & 0\;\;\;\;\; &\cdots\;\;\;\;\; &\prod\limits_{i=1}^{h-1}(hk-i)\,t^{hk-h}\end{pmatrix}\in {\mathbb F}_q^{h\times hk}.\end{equation}

According to (2·5), the osculating space of order $h-1$ at $\nu_{hk,q}(0,1)$ is spanned by the rows of the matrix

(3·10)

Finally, for each $\alpha\in\Lambda_{h,q}$ , define $B_{\alpha}$ as the matrix whose rows are the vectors representing the points $P_{\alpha},P_{\alpha}^{\Psi},\ldots,P_{\alpha}^{\Psi^{h-1}}$ , namely

\begin{equation*}B_{\alpha} =\begin{pmatrix}1\;\;\;\;\; & \alpha\;\;\;\;\; & \alpha^{2}\;\;\;\;\; & \cdots\;\;\;\;\; & \alpha^{hk-1} \\[4pt]1\;\;\;\;\; & \alpha^{q}\;\;\;\;\; & \alpha^{2q}\;\;\;\;\; & \cdots\;\;\;\;\; & \alpha^{(hk-1)q} \\[4pt]1\;\;\;\;\; & \alpha^{q^{2}}\;\;\;\;\; & \alpha^{2q^{2}}\;\;\;\;\; & \cdots\;\;\;\;\; & \alpha^{(hk-1)q^{2}} \\[4pt]\vdots\;\;\;\;\; & \vdots\;\;\;\;\; & \vdots\;\;\;\;\; & \ddots\;\;\;\;\; & \vdots \\[4pt]1\;\;\;\;\; & \alpha^{q^{h-1}}\;\;\;\;\; & \alpha^{2q^{h-1}}\;\;\;\;\; & \cdots\;\;\;\;\; & \alpha^{(hk-1)q^{\,h-1}}\end{pmatrix}\in {\mathbb F}_{q^h}^{h\times hk}.\end{equation*}

The row space of $B_{\alpha}$ spans the subspace $X^*(P_{\alpha})$ associated with the orbit of $P_{\alpha}$ under $\Psi$ . To prove the assertion, it is enough to show that for any $j\in\{1,\ldots,k\}$ , if we take j pairwise distinct elements $t_1,\ldots,t_{j}\in{\mathbb F}_q \cup \{\infty\}$ and $k-j$ pairwise distinct elements $\alpha_1,\ldots,\alpha_{k-j}\in \Lambda_{h,q}$ , then the block matrix

\begin{equation*}N \,:\!=\,\begin{pmatrix}A_{t_1}\\[2pt]\vdots\\[2pt]A_{t_{j}}\\[2pt]B_{\alpha_1}\\[2pt]\vdots\\[2pt]B_{\alpha_{k-j}}\end{pmatrix}\in {\mathbb F}_{q^h}^{hk\times hk}\end{equation*}

is nonsingular. When $j=k$ , only the matrices $A_{t_i}$ appear in N.

Assume, for contradiction, that N is singular. Then there exists a nonzero vector

\begin{equation*}\mathbf{c}=(c_0,\ldots,c_{hk-1})\in{\mathbb F}_{q^h}^{hk}\end{equation*}

such that $N\mathbf{c}^{\top}=\mathbf{0}$ . This means that

\begin{equation*}A_{t_i}\mathbf{c}^{\top}=\mathbf{0}\quad\text{for all }i\in\{1,\ldots,j\},\qquad B_{\alpha_r}\mathbf{c}^{\top}=\mathbf{0}\quad\text{for all }r\in\{1,\ldots,k-j\}.\end{equation*}

Define the polynomial

\begin{equation*}f(Z)=\sum_{i=0}^{hk-1}c_i Z^i\in{\mathbb F}_{q^h}[Z].\end{equation*}

From $B_{\alpha_r}\mathbf{c}^{\top}=\mathbf{0}$ we deduce that

\begin{equation*}f\big(\alpha_r^{q^s}\big)=0,\qquad r\in\{1,\dots,k-j\},\ s\in\{0,1,\dots,h-1\},\end{equation*}

since the s-th row of $B_{\alpha_r}$ corresponds to the evaluation of f at $\alpha_r^{q^s}$ . Because ${\mathbb F}_q(\alpha_r)={\mathbb F}_{q^h}$ , the conjugates $\alpha_r,\alpha_r^{q},\ldots,\alpha_r^{q^{h-1}}$ are h distinct elements of ${\mathbb F}_{q^h}$ . Hence each $\alpha_r$ contributes h distinct simple zeros of f.

From $A_{t_i}\mathbf{c}^{\top}=\mathbf{0}$ , if $t_i\neq\infty$ we obtain

\begin{equation*}f^{(\ell)}(t_i)=0,\quad \text{for all }\ell\in\{0,1,\dots,h-1\},\end{equation*}

where $f^{(\ell)}$ denotes the $\ell$ -th derivative of f. Thus each $t_i$ is a root of multiplicity at least h. In total, f has at least $j\cdot h$ zeros (counted with multiplicity) coming from the $t_i$ , and $(k-j)\cdot h$ distinct simple zeros coming from the $\alpha_r$ . Hence f has at least hk zeros in ${\mathbb F}_{q^h}$ counting multiplicities. Since $\deg f\le hk-1$ , this is impossible for a nonzero polynomial. Therefore f must be the zero polynomial, implying $\mathbf{c}=\mathbf{0}$ . Hence N is nonsingular. Finally, if one of the parameters $t_i=\infty$ , then the last h coordinates of $\mathbf{c}$ vanish, that is, $c_{hk-h}=c_{hk-h+1}=\cdots=c_{hk-1}=0$ . In this case f has degree at most $hk-h-1$ , and the same argument as above applies verbatim, yielding the desired contradiction.

We now compare the size of the pseudo-arc ${\mathcal P}_{h,k,q} \cup\ \mathcal{O}_{h-1}$ with the upper bound in (1·1). For fixed $h$ , we have

\begin{equation*}|\Lambda_{h,q}|=\mathcal{O}(q^h)\quad \text{as } q\to\infty.\end{equation*}

Therefore, the pseudo-arc ${\mathcal P}_{h,k,q} \cup\ \mathcal{O}_{h-1}$ has size

\begin{equation*}|\Lambda_{h,q}|+q+1=\mathcal{O}(q^h).\end{equation*}

In particular, for fixed $h$ and $k$ , its size has the same asymptotic order as ${q^h}$ as $q \to \infty $ . Thus the order of magnitude of the provided constructions matches with the order of magnitude of the upper bound in (1·1), also in the non-Desarguesian setting considered here.

3·4. Pseudo-arcs and quadrics

Quadrics are intimately connected with arcs. For instance, in [ Reference Glynn20 ] it is proved that every arc in $\mathrm{PG}(k-1,q)$ having size $2k-1$ is contained in the intersection of a unique projective space S of quadrics of dimension $\binom{k-1}2-1$ that does not contain a quadric that is the union of two hyperplanes. Moreover, a normal rational curve in $\mathrm{PG}(k-1,q)$ is the complete intersection of such a space S of quadrics. We refer to [ Reference Glynn20 ] for further results of this type. In the following, we investigate this relationship in the context of pseudo-arcs. A pseudo-arc is said to be an intersection of quadrics if there exists a family of quadrics whose intersection contains each of its subspaces. It is called a complete intersection of quadrics if that intersection contains no further points besides those lying in the union of its subspaces. By Proposition 2·7, we see that if an arc ${\mathcal A}$ in $\Theta \simeq \mathrm{PG}(k-1, q^h)$ is intersection of a family of quadrics, then the members of the associated pseudo-arc

\begin{equation*}X({\mathcal A}) = \{X(P) \;:\; P \in {\mathcal A}\}\end{equation*}

are still contained in the intersection of a suitable family of quadrics. Moreover, $X({\mathcal A})$ is a complete intersection of quadrics if ${\mathcal A}$ does.

By [ Reference Glynn20 , theorem 3·3], a normal rational curve of $\mathrm{PG}(k-1,q)$ is the complete intersection of a space of quadrics not containing any hyperplane-pair. More precisely, consider the normal rational curve

\begin{equation*}\mathcal{N}_{k,q}=\{\langle (1,t,t^2,\ldots,t^{k-1})\rangle_{{\mathbb F}_{q}} \,:\, t\in{\mathbb F}_q\}\cup\{\langle(0,\ldots,0,1)\rangle_{{\mathbb F}_{q}}\}.\end{equation*}

The space of quadrics whose complete intersection is $\mathcal{N}_{k,q}$ is generated by the quadrics

(3·11) \begin{equation} {\mathcal Q}_{i,j}\colon\; x_i x_j-x_{i+1}x_{j-1}=0,\qquad 1\le i \leq j-2, \ 2 \leq j\le k,\end{equation}

of $\mathrm{PG}(k-1,q)$ which span a vector space of dimension $\binom{k-1}{2}$ . None of these quadrics is a union of two hyperplanes, and their intersection coincides exactly with the normal rational curve. Clearly, we can apply this description to a normal rational curve $\mathcal{N}_{k,q^h}$ of $\Theta \simeq \mathrm{PG}(k-1,q^h)$ . By Proposition 3·13, the Desarguesian pseudo-arc $X(\mathcal{N}_{k,q^h})$ of $\mathrm{PG}(hk-1,q)$ , associated with $\mathcal{N}_{k,q^h}$ , is a complete intersection of quadrics as well.

Now, we investigate the situation for the pseudo-arc ${\mathcal P}_{h,k,q}$ constructed from the imaginary points of a normal rational curve as in Theorem 3·4. We prove that, despite the fact that it also originates from points of a normal rational curve, its geometric behaviour is fundamentally different: no quadric of the ambient projective space contains all of its elements. Consequently, this pseudo-arc is not an intersection of quadrics.

We consider the same setting as in Section 3·1, with $\Sigma\cong\mathrm{PG}(hk-1,q)$ , $\Psi$ and the normal rational curve $\mathcal{N}_{hk,q^h}$ as defined therein.

Proof. Assume, by contradiction, that there exists a quadric $\mathcal{Q}$ of $\mathrm{PG}(hk-1,q)$ containing all the $(h-1)$ -dimensional projective spaces

\begin{equation*}X(P_{\alpha}) = X^*(P_{\alpha})\cap \Sigma,\quad \alpha\in\Lambda_{h,q},\end{equation*}

where $P_{\alpha}$ is as in (3·6). Let $\mathcal{Q}^*$ denote the extension of $\mathcal{Q}$ to $\mathrm{PG}(hk-1,q^h)$ , obtained by viewing the defining homogeneous quadratic equation of $\mathcal{Q}$ over ${\mathbb F}_{q^h}$ . Since $\mathcal{Q}$ contains $X(P_{\alpha})$ , it follows that $\mathcal{Q}^*$ contains $X^*(P_{\alpha})$ . In particular, $\mathcal{Q}^*$ contains the points $P_{\alpha},P_{\alpha}^{\Psi},\ldots,P_{\alpha}^{\Psi^{h-1}}$ for every $\alpha\in\Lambda_{h,q}$ . These are pairwise distinct points of the normal rational curve $\mathcal{N}_{hk,q^h}$ . Therefore, $\mathcal{Q}^*$ contains the sub-arc

\begin{equation*}\mathcal{A}_0\,:\!=\,\big\{P_{\alpha}^{\Psi^i} \,:\, \alpha\in\Lambda_{h,q},\ i=0,\ldots,h-1\big\}\subseteq \mathcal{N}_{hk,q^h},\end{equation*}

whose size is

\begin{equation*}|\mathcal{A}_0|= h|\Lambda_{h,q}|\ge q^h-\sum_{\substack{d\mid h\\ d \lt h}} q^d\ge q^h-q^{h-1}.\end{equation*}

In particular, since $q\ge hk$ , then $|\mathcal{A}_0|\ge 2hk-1$ . Hence, by [ Reference Glynn20 , theorem 3·3], the quadric $\mathcal{Q}^*$ vanishes on the whole normal rational curve $\mathcal{N}_{hk,q^h}$ and must belong to the space of quadrics defining $\mathcal{N}_{hk,q^h}$ , namely the space generated by the quadrics of $\mathrm{PG}(hk-1,q^h)$

\begin{equation*}\mathcal{Q}_{i,j}^*\,:\,x_i x_j-x_{i+1}x_{j-1}=0,\qquad 1 \le i \le j-2,\ 3\le j\le hk,\end{equation*}

cf. (3·11). In particular, we may write

\begin{equation*}\mathcal{Q}^*\,:\,\sum_{\substack{1 \le i \le j-2\\ 3 \le j \le hk}}\lambda_{i,j}\,(x_i x_j-x_{i+1}x_{j-1})=0,\end{equation*}

for some coefficients $\lambda_{i,j}\in{\mathbb F}_{q^h}$ , not all zero. It can then be checked by direct computation that the bilinear form

\begin{equation*}B^*(\mathbf{x},\mathbf{y})=\sum_{i,j}\gamma_{i,j}x_i y_j,\end{equation*}

with $\mathbf{x}=(x_1,\ldots,x_{hk})$ and $\mathbf{y}=(y_1,\ldots,y_{hk})$ , is the bilinear form associated with $\mathcal{Q}^*$ , where the coefficients $\gamma_{i,j}$ are given by

\begin{equation*}\begin{cases}\gamma_{1,1}=\gamma_{1,2}=\gamma_{hk-1,hk}=\gamma_{hk,hk}=0,\\\gamma_{1,j}=\lambda_{1,j}, & \text{for all } j\ge 3,\\\gamma_{i,hk}=\lambda_{i,hk}, & \text{for all } i\le hk-2,\\\gamma_{i,i}=-2\lambda_{i-1,i+1}, & \text{for all } 2\le i\le hk-1,\\\gamma_{i,i+1}=-\lambda_{i-1,i+2}, & \text{for all } 2\le i\le hk-2,\\\gamma_{j,i}=\gamma_{i,j}=\lambda_{i,j}-\lambda_{i-1,j+1},& \text{for all } 2\le i\le j-2,\ 4\le j\le hk-1.\end{cases}\end{equation*}

Since the coefficients $\lambda_{i,j}$ are not all zero, it follows from the above expressions that the bilinear form $B^*$ is not identically zero. Moreover, since $X^*(P_\alpha)\subseteq \mathcal{Q}^*$ for every $\alpha\in\Lambda_{h,q}$ , and since $P_\alpha,P_\alpha^{\Psi},\ldots,P_\alpha^{\Psi^{h-1}}$ belong to $X^*(P_\alpha)$ , it follows from [ Reference Hirschfeld and Thas26 , lemma 1·5] that

(3·12) \begin{equation} B^*\big(P_\alpha^{\Psi^i},P_\alpha^{\Psi^{i+1}}\big)=0\end{equation}

for every $\alpha\in\Lambda_{h,q}$ and $i=0,\ldots,h-1$ .

Now, let $\mathbf{Z}=(1,Z,\ldots,Z^{hk-1})$ , and consider the polynomial

\begin{equation*}f(Z)\,:\!=\,B^*(\mathbf{Z},\mathbf{Z}^q)=\sum_{i,j}\gamma_{i,j}Z^{i-1}Z^{q(j-1)}.\end{equation*}

Since $q\ge hk$ and $B^*$ is non-zero, f(Z) is a non-zero polynomial over ${\mathbb F}_{q^h}$ of degree at most $(hk-1)(q+1)$ . By (3·12), for every $\alpha\in\Lambda_{h,q}$ the elements $\alpha,\alpha^q,\ldots,\alpha^{q^{h-1}}$ are zeros of f(Z). Hence, f(Z) has at least

\begin{equation*}h|\Lambda_{h,q}|\ge q^h-q^{h-1}\end{equation*}

distinct zeros. In particular, for $q\ge hk+1$ this number is strictly larger than $\deg(f)\le$ $(hk-1)(q+1)$ , which contradicts the degree bound. Therefore, no such quadric $\mathcal{Q}$ exists.

4·1. Basics on additive Hamming metric codes

We begin by recalling some basic notions of Hamming-metric codes that will be needed in the following. For further background, we refer the reader to [ Reference Ball3 ].

Let n be a positive integer and let ${\mathbb F}$ denote a finite field. The Hamming distance on ${\mathbb F}^n$ is defined by

\begin{equation*}\mathrm{d}_{\mathrm{H}}(\mathbf{x},\mathbf{y}) \,:\!=\, \lvert \{ i \,:\, x_i \neq y_i \} \rvert,\end{equation*}

for all vectors $\mathbf{x}=(x_1,\ldots,x_n)$ and $\mathbf{y}=(y_1,\ldots,y_n)$ in ${\mathbb F}^n$ . A Hamming-metric code is simply a subset of the metric space $({\mathbb F}^n,\mathrm{d}_{\mathrm{H}})$ . A code $\mathcal{C}$ is called additive if it is closed under vector addition, i.e., if $\mathbf{x},\mathbf{y} \in \mathcal{C}$ implies $\mathbf{x}+\mathbf{y} \in \mathcal{C}$ . Suppose now that the alphabet is ${\mathbb F} = {\mathbb F}_{q^h}$ , the finite field with $q^h$ elements. Viewing ${\mathbb F}_{q^h}$ as a vector space over its subfield ${\mathbb F}_q$ , we say that a code $\mathcal{C} \subseteq {\mathbb F}_{q^h}^n$ is ${\mathbb F}_q$ -linear if it is an ${\mathbb F}_q$ -subspace of $({\mathbb F}_{q^h}^n, \mathrm{d}_{\mathrm{H}})$ .

We write that $\mathcal{C}$ is an ${\mathbb F}_q$ -linear $(n,q^k,d)_{q^h}$ code if $\mathcal{C}$ is an ${\mathbb F}_q$ -linear Hamming-metric code in ${\mathbb F}_{q^h}^n$ , $\dim_{{\mathbb F}_q}(\mathcal{C}) = k$ , and d is the minimum distance:

\begin{equation*} d = \mathrm{d}_{\mathrm{H}}(\mathcal{C}) \,:\!=\, \min\{ \mathrm{d}_{\mathrm{H}}(\mathbf{c}_1, \mathbf{c}_2) \,:\, \mathbf{c}_1, \mathbf{c}_2 \in \mathcal{C},\ \mathbf{c}_1 \ne \mathbf{c}_2 \}. \end{equation*}

If the minimum distance is not known or not relevant, we simply denote $\mathcal{C}$ as an ${\mathbb F}_q$ -linear $(n,q^k)_{q^h}$ code. Note that every additive code is ${\mathbb F}_p$ -linear, and every ${\mathbb F}_q$ -linear code is additive. When $\mathcal{C}$ is ${\mathbb F}_{q^h}$ -linear, we refer to it simply as a linear code. The Hamming weight of a vector $\mathbf{x} = (x_1,\ldots,x_n) \in {\mathbb F}_{q^h}^n$ is defined as

\begin{equation*}\mathrm{w}_{\mathrm{H}}(\mathbf{x}) \,:\!=\, \mathrm{d}_{\mathrm{H}}(\mathbf{x}, \mathbf{0}) = \lvert \{ i \,:\, x_i \neq 0 \} \rvert.\end{equation*}

Since $\mathrm{d}_{\mathrm{H}}(\mathbf{x},\mathbf{y}) = \mathrm{w}_{\mathrm{H}}(\mathbf{x} - \mathbf{y})$ for all $\mathbf{x}, \mathbf{y} \in {\mathbb F}_{q^h}^n$ , the minimum distance of an additive code $\mathcal{C}$ can be equivalently expressed as $\mathrm{d}_{\mathrm{H}}(\mathcal{C}) = \min\{ \mathrm{w}_{\mathrm{H}}(\mathbf{c}) \,:\, \mathbf{c} \in \mathcal{C} \setminus \{\mathbf{0}\} \},$ see e.g. [ Reference Ball, Gamboa and Lavrauw6 , lemma 3]. The fundamental trade-off between the parameters of a Hamming-metric code is expressed by the classical Singleton bound: if $\mathcal{C}$ is an ${\mathbb F}_q$ -linear $(n,q^k,d)_{q^h}$ code, then

\begin{equation*} q^k \;\leq\; q^{h(n-d+1)}. \end{equation*}

Codes that attain equality in the Singleton bound are called maximum distance separable codes, or simply MDS codes.

Definition 4·1. Let $\mathcal{C}$ and $\mathcal{C}'$ be ${\mathbb F}_q$ -linear Hamming-metric codes in ${\mathbb F}_{q^h}^n$ . We say that $\mathcal{C}$ and $\mathcal{C}'$ are ${\mathbb F}_q$ -linearly equivalent if there exist ${\mathbb F}_q$ -linear automorphism $\phi_1,\ldots,\phi_n$ of ${\mathbb F}_{q^h}$ and a permutation $\theta$ of $\{1,\ldots,n\}$ such that

\begin{equation*} \mathcal{C}' = \{ (\phi_1(x_{\theta(1)}),\ldots,\phi_n(x_{\theta(n)})) \,:\, (x_1,\ldots,x_n) \in \mathcal{C} \}. \end{equation*}

Codes that are ${\mathbb F}_q$ -linear equivalent clearly have the same size and the same minimum distance. Note that this notion of additive equivalence includes the monomial equivalence, where each $\phi_i$ is of the form $ y \mapsto \alpha_i y, $ with $\alpha_i \in {\mathbb F}_{q^h}^*, $ in which case the codes are also called monomial (or linear) equivalent, see e.g. [ Reference Huffman and Pless27 ].

Finally, we recall that in the sequel we only consider Hamming-metric codes that are nondegenerate, meaning that the code cannot be isometrically embedded into a smaller ambient space.

4·2. The geometry of additive Hamming metric codes

There is a well-known geometric interpretation of codes endowed with the Hamming metric. A linear $(n,q^k,d)_q$ code $\mathcal{C}$ corresponds to a multiset of points in $\mathrm{PG}(k-1,q)$ , and conversely, see e.g. [ Reference Tsfasman and Vlăduţ49 ]. In [ Reference Ball, Gamboa and Lavrauw6 ], this correspondence was extended to additive codes over finite fields, see also [ Reference Adriaensen and Ball1 ]. Below we briefly recall this viewpoint.

A generator matrix of a $(n, q^{hk}, d)_{q^h}$ code ${\mathcal C}$ , which is linear over ${\mathbb F}_q$ , is an $hk \times n$ matrix with entries from ${\mathbb F}_{qh}$ , whose row space over ${\mathbb F}_q$ is ${\mathcal C}$ . Fix an ${\mathbb F}_q$ -basis $\{\omega_1,\ldots,\omega_h\}$ of ${\mathbb F}_{q^h}$ and let $\boldsymbol{\omega}= (\omega_1,\ldots,\omega_h)^{\top}$ . Then the j-th column of G is of the form $G_j \boldsymbol{\omega}$ , for some $G_j \in {\mathbb F}_q^{hk \times h}$ . Define $\pi_j$ as the subspace of $\mathrm{PG}(hk-1,q)$ corresponding to the column space of $G_j$ , for each $j \in \{1,\ldots, n\}$ . Each $\pi_j$ has projective dimension at most $h-1$ . Also the subspace $\pi_j$ is independent of the chosen ${\mathbb F}_q$ -basis of ${\mathbb F}_{q^h}$ , see [ Reference Adriaensen and Ball1 , remark 3·1]. Therefore, we can consider the multiset $\mathcal{U}=\{\pi_1,\ldots,\pi_n\}$ . Conversely, given a multi-set of subspaces of $\mathrm{PG}(hk-1, q)$ of projective dimension at most $h-1$ , once that an ${\mathbb F}_q$ -basis $\{\omega_1,\ldots,\omega_h\}$ of ${\mathbb F}_{q^h}$ is fixed, an additive code ${\mathcal C}$ can be constructed by reversing the above process.

Note that different choices of the generator matrix G yield different representatives of the same multisets of subspaces. Precisely, if G ^′ is another generator matrix for $\mathcal{C}$ , then $G' = MG$ for some $M \in {\mathrm{GL}}(hk,q)$ , and the multisets of subspaces defined by G and G ^′ lie in the same orbit under $\mathrm{PGL}(hk,q)$ . The following holds.

For the case $h=1$ , the correspondence reduces to the classical connection between linear Hamming-metric codes and multiset of points in a projective space.

We now turn our attention to MDS codes. It is a classical result that linear MDS $(n,q^k)_q$ codes correspond to arcs in $\mathrm{PG}(k-1,q)$ . Using the geometric description of additive codes, this correspondence can be extended to provide the geometric counterpart of additive MDS codes.

4·3. The pseudo-arc $\mathcal{P}_{h,k,q}$ and additive MDS codes

In view of Theorem 4·5, the pseudo-arc $\mathcal{P}_{h,k,q}$ constructed in Theorem 3·4 correspond to additive MDS codes. In this subsection, we show that these additive MDS codes coincide precisely with the family of additive MDS codes recently introduced in [ Reference Neri and Santonastaso33 ]. Exploiting the geometric properties established for these pseudo-arcs, we then prove that the corresponding codes are not equivalent to any linear MDS code.

We start by recalling the additive MDS codes introduced in [ Reference Neri and Santonastaso33 ].

Definition 4·6. Let $\boldsymbol{\alpha} = (\alpha_1,\ldots,\alpha_n)$ , where $\alpha_1,\ldots,\alpha_n$ are distinct elements of $\Lambda_{h,q}$ . For any integer $1 \leq k \lt n$ , define the code

(4·1) \begin{equation} \mathcal{S}_{h,k,q}(\boldsymbol{\alpha}) \,:\!=\, \{ (f(\alpha_1),\ldots,f(\alpha_n)) \,:\, f(x) \in {\mathbb F}_q[x],\ \deg(f) \lt hk \} \subseteq {\mathbb F}_{q^h}^n.\end{equation}

The codes $\mathcal{S}_{h,k,q}(\boldsymbol{\alpha})$ can be regarded as the sublinear analogue of Reed–Solomon codes. Indeed, they are ${\mathbb F}_q$ -linear MDS codes over ${\mathbb F}_{q^h}$ , and can be obtained as subcodes of classical Reed–Solomon codes over ${\mathbb F}_{q^h}$ of dimension hk, evaluated on subsets of $\Lambda_{h,q}$ . As proved in [ Reference Neri and Santonastaso33 , theorem 5·17], the code $\mathcal{S}_{h,k,q}(\boldsymbol{\alpha})$ is an ${\mathbb F}_q$ -linear $(n,q^{hk},n-k+1\,)_{q^h}$ code, and hence an ${\mathbb F}_q$ -linear MDS code in ${\mathbb F}_{q^h}^{n}$ . The maximum length occurs when $n = |\Lambda_{h,q}|$ . Note that a generator matrix of $\mathcal{S}_{h,k,q}(\boldsymbol{\alpha})$ is given by

(4·2) \begin{equation} G_{h,k,q}(\boldsymbol{\alpha})=\begin{bmatrix}1\;\;\;\;\; & 1\;\;\;\;\; & \cdots\;\;\;\;\; & 1\\\alpha_1\;\;\;\;\; & \alpha_2\;\;\;\;\; & \cdots\;\;\;\;\; & \alpha_n\\\alpha_1^{2}\;\;\;\;\; & \alpha_2^{2}\;\;\;\;\; & \cdots\;\;\;\;\; & \alpha_n^{2}\\\vdots\;\;\;\;\; & \vdots\;\;\;\;\; & \ddots\;\;\;\;\; & \vdots\\\alpha_1^{\,hk-1}\;\;\;\;\; & \alpha_2^{\,hk-1}\;\;\;\;\; & \cdots\;\;\;\;\; & \alpha_n^{\,hk-1}\end{bmatrix} \in {\mathbb F}_{q^h}^{hk \times n}.\end{equation}

Proof. A generator matrix for $\mathcal{S}_{h,k,q}(\boldsymbol{\alpha})$ is $G = G_{h,k,q}(\boldsymbol{\alpha})$ as in (4·2). Let $\omega$ be a normal element of the extension ${\mathbb F}_{q^h}/{\mathbb F}_q$ , and set $\boldsymbol{\omega} = (\omega, \omega^q, \ldots, \omega^{q^{h-1}})^{\top}$ . The j-th column of G can be written as

(4·3) \begin{equation} \begin{pmatrix}1 \\\alpha_j \\\vdots \\\alpha_j^{hk-1}\end{pmatrix}= G_j \boldsymbol{\omega},\end{equation}

for some $G_j \in {\mathbb F}_q^{hk \times h}$ . Keeping the notation of Section 3·1, recall that

\begin{equation*}X^{*}(P_{\alpha_j}) = \big\langle P_{\alpha_j}, P_{\alpha_j}^{\Psi}, \ldots, P_{\alpha_j}^{\Psi^{h-1}} \big\rangle_{{\mathbb F}_{q^h}},\end{equation*}

where $P_{\alpha_j}$ is as in (3·6) and note that

\begin{equation*}P_{\alpha_j}^{\Psi^i} = \big\langle \big(1, \alpha_j^{q^i}, \big(\alpha_j^2\big)^{q^i}, \ldots, \big(\alpha_j^{hk-1}\big)^{q^i}\big) \big\rangle_{{\mathbb F}_{q^h}}.\end{equation*}

If $G_j^1,\ldots,G_j^h$ denote the columns of $G_j$ , by (4·3), we have

\begin{equation*}P_{\alpha_j} \in \big\langle \big(G_j^1\big)^{\top},\ldots,\big(G_j^h\big)^{\top} \big\rangle_{{\mathbb F}_{q^h}}.\end{equation*}

Moreover, for every i,

\begin{equation*}\begin{pmatrix}1 \\\alpha_j^{q^i} \\\big(\alpha_j^2\big)^{q^i} \\\vdots \\\big(\alpha_j^{hk-1}\big)^{q^i}\end{pmatrix}= G_j \begin{pmatrix} \omega^{q^i} \\ \omega^{q^{i+1}} \\ \vdots \\ \omega^{q^{h-1}} \\ \omega \\ \vdots \\ \omega^{q^{i-1}}\end{pmatrix}\!,\end{equation*}

so that $P_{\alpha_j}^{\Psi^i} \in \langle (G_j^1)^{\top},\ldots,(G_j^h)^{\top} \rangle_{{\mathbb F}_{q^h}}$ as well. Therefore,

\begin{equation*}X(P_{\alpha_j}) = X^{*}(P_{\alpha_j}) \cap \Sigma = \mathrm{PG}(U_j,{\mathbb F}_q),\end{equation*}

where $U_j$ is the ${\mathbb F}_q$ -column space generated by the columns of $G_j$ and so a pseudo-arc associated with $\mathcal{S}_{h,k,q}(\boldsymbol{\alpha})$ is ${\mathcal P}_{h,k,q}$ provided in Theorem 3·4.

The question of determining whether the codes $\mathcal{S}_{h,k,q}(\boldsymbol{\alpha})$ are equivalent to linear MDS codes remained open in [ Reference Neri and Santonastaso33 ]. In what follows, we answer this question in the negative by adopting a geometric approach. A geometric criterion for determining when an additive code is ${\mathbb F}_q$ -equivalent to a linear code is recalled below.

Since, by Theorem 3·10, the pseudo-arc $\mathcal{P}_{h,k,q}$ is non-Desarguesian, we obtain the following result.

4·4. Extending the code $\mathcal{S}_{h,k,q}(\boldsymbol{\alpha})$

As shown in Section 3·3, the pseudo-arc $\mathcal{P}_{h,k,q}$ can be further extended to a larger pseudo-arc. An immediate consequence on the coding-theoretic side is that the corresponding codes $\mathcal{S}_{h,k,q}(\boldsymbol{\alpha})$ can be extended to longer additive MDS codes. In this section, we provide an explicit description of these extended codes.

Definition 4·10. Let $\boldsymbol{\alpha} = (\alpha_1,\ldots,\alpha_n)$ , where $\alpha_1,\ldots,\alpha_n$ are distinct elements of $\Lambda_{h,q}$ , and let $\mathbf{t}=(t_1,\ldots,t_m)$ , where $t_1,\ldots,t_m$ are distinct elements of ${\mathbb F}_q$ . Let $\omega$ be a normal element of ${\mathbb F}_{q^h}$ over ${\mathbb F}_q$ . For any integer $1 \le k \lt n$ , define the code

(4·4) \begin{equation} \begin{aligned}\mathcal{S}_{h,k,q}(\boldsymbol{\alpha},\mathbf{t},\infty) \,:\!=\, \Bigg\{(&f(\alpha_1),\ldots,f(\alpha_n), \sum_{i=0}^{h-1} f^{(i)}(t_1)\,\omega^{q^i}, \ldots, \sum_{i=0}^{h-1} f^{(i)}(t_m)\,\omega^{q^i}, f(\overline{\infty})) \\ & :\, f(x)\in{\mathbb F}_q[x],\ \deg(f) \lt hk \,\Bigg\} \subseteq {\mathbb F}_{q^h}^{\,n+m+1}.\end{aligned}\end{equation}

where, if $f(x)=\sum_{j=0}^{hk-1} f_j x^j$ , we define $f(\overline{\infty}) \,:\!=\, \sum_{i=0}^{h-1} f_{hk-h+i}\,\omega^{q^i}$ .

The code $\mathcal{S}_{h,k,q}(\boldsymbol{\alpha},\mathbf{t},\infty)$ extends $\mathcal{S}_{h,k,q}(\boldsymbol{\alpha})$ by adding evaluation points corresponding to derivatives and to the point at infinity. We prove that the resulting code is still an additive MDS code by showing that it is precisely the code associated with the extended pseudo-arc $\mathcal{P}_{h,k,q} \cup \mathcal{O}_{h-1}$ as defined in Theorem 3·11.

Proof. By considering as an ${\mathbb F}_q$ -basis of the space of polynomials in ${\mathbb F}_q[x]$ of degree less than hk the set $\{1,x,x^2,\ldots,x^{hk-1}\},$ we can explicitly describe a generator matrix of $\mathcal{S}_{h,k,q}(\boldsymbol{\alpha},\mathbf{t},\infty)$ . Set $\boldsymbol{\omega} = (\omega, \omega^q, \ldots, \omega^{q^{h-1}})^{\top}$ . More precisely, a generator matrix is given by

\begin{equation*}G = \bigl( G_{h,k,q}(\boldsymbol{\alpha}) \;\big|\; H_1 \;\big|\; \cdots \;\big|\; H_m \;\big|\; H_{\infty} \bigr),\end{equation*}

where $G_{h,k,q}(\boldsymbol{\alpha})$ is as in (4·2), $H_i = A_{t_i}^{\top}\boldsymbol{\omega}$ for $i=1,\ldots,m$ , and $H_{\infty} = A_{\infty}^{\top}\boldsymbol{\omega}$ . Here $A_{t_i}$ and $A_{\infty}$ are defined in (3·9) and (3·10), respectively. Hence, the set of subspaces obtained from the generator matrix G of $\mathcal{S}_{h,k,q}(\boldsymbol{\alpha},\mathbf{t},\infty)$ is $\mathcal{P}_{h,k,q} \cup \mathcal{O}_{h-1}$ . The assertion follows by Theorem 3·11 and Theorem 4·5.

We remark that the maximum length of the code $\mathcal{S}_{h,k,q}(\boldsymbol{\alpha},\mathbf{t},\infty)$ is

\begin{equation*}|\Lambda_{h,q}| + q + 1,\end{equation*}

and that, if $q \ge hk-1$ , it is not ${\mathbb F}_q$ -equivalent to any linear MDS code in ${\mathbb F}_{q^h}^n$ , as immediate consequence of Proposition 4·9.

4·5. Further comments

For small parameters, additive MDS codes that are not equivalent to linear MDS codes have been classified in [ Reference Ball, Gamboa and Lavrauw6 , section 6]. In particular, the authors have proved that the longest $(n, 4^6, n-2)_{4^2}$ additive MDS code that is not equivalent to a linear MDS code has parameters $(11, 4^6,9)_{4^2}$ . Moreover, the corresponding pseudo-arc ${\mathcal P}$ is unique up to a collineation of $\mathrm{PG}(5, 4)$ and can be described as follows. Let w be a primitive element of ${\mathbb F}_{4^2}$ such that $w^4 = w+1$ . Then $e = w^5$ is a primitive element of ${\mathbb F}_{4}$ satisfying $e^2 = e+1$ and ${\mathcal P}$ consists of the lines:

\begin{align*} & \ell_1 = \langle (1,0,0,0,0,0), (0,1,0,0,0,0) \rangle_{{\mathbb F}_4}, && \ell_2 = \langle (1,0,e,e,e,1), (0,1,e,1,1,1) \rangle_{{\mathbb F}_4}, \\ & \ell_3 = \langle (1,0,1,0,1,0), (0,1,0,1,0,1) \rangle_{{\mathbb F}_4}, && \ell_{4} = \langle (1,0,0,e,1,e), (0,1,e,e^2,e,0) \rangle_{{\mathbb F}_4}, \\ & \ell_5 = \langle (1,0,e,e^2,0,1), (0,1,e^2,e^2,1,e^2) \rangle_{{\mathbb F}_4}, && \ell_6 = \langle (0,0,1,0,0,0), (0,0,0,1,0,0) \rangle_{{\mathbb F}_4}, \\ & \ell_7 = \langle (0,0,0,0,1,0), (0,0,0,0,0,1) \rangle_{{\mathbb F}_4}, && \ell_8 = \langle (1,0,0,1,0,e), (0,1,1,e^2,e,e^2) \rangle_{{\mathbb F}_4}, \\ & \ell_9 = \langle (1,0,1,e^2,e^2,1), (0,1,1,1,0,e) \rangle_{{\mathbb F}_4}, && \ell_{10} = \langle (1,0,1,e,e,0), (0,1,e^2,0,e,e) \rangle_{{\mathbb F}_4}, \\ & \ell_{11} = \langle (1,0,e,0,e,e), (0,1,1,e,e^2,e) \rangle_{{\mathbb F}_4}.\end{align*}

Actually, the pseudo-arc ${\mathcal{P}}$ can be described as the pseudo-arc ${\mathcal P}_{h,k,q}\cup \mathcal{O}_{h-1}$ in Theorem 3·11, with $k = 3$ , $h = 2$ , and $q = 4$ . In this case, the size of ${{\mathcal{P}}_{2,3,4}} \cup {{\mathcal{O}}_1}$ is

\begin{equation*}|\Lambda_{h,q}|+q+1=|\Lambda_{2,4}|+5=\frac{4^2-4}{2}+5=11,\end{equation*}

since $h = 2$ is prime and hence $|\Lambda_{h,q}|=\frac{q^h-q}{h}$ , and ${\mathcal P}_{2,3,4}\cup \mathcal{O}_{1}$ coincides with ${\mathcal{P}}$ . Indeed, embed $\mathrm{PG}(5, 4)$ in $\mathrm{PG}(5, 4^2)$ and consider the set ${\mathcal N}$ formed by the points

\begin{align*} & L_1 = \langle (1,0,0,0,0,0)\rangle_{{\mathbb F}_{16}}, && L_2 = \langle (1,1,0,e^2,e^2,0) \rangle_{{\mathbb F}_{16}}, \\ & L_3 = \langle (1,e,1,e,1,e) \rangle_{{\mathbb F}_{16}}, && L_4 = \langle (1,e^2,1,0,0,e) \rangle_{{\mathbb F}_{16}}, \\ & L_5 = \langle (0,1,e^2,e^2,1,e^2) \rangle_{{\mathbb F}_{16}}, && L_6 = \langle (0,0,1,w^4,0,0) \rangle_{{\mathbb F}_{16}}, \\ & L_6^{\Psi} = \langle (0,0,1,w^6,0,0) \rangle_{{\mathbb F}_{16}}, && L_7 = \langle (0,0,0,0,1,w^3) \rangle_{{\mathbb F}_{16}}, \\ & L_7^{\Psi} = \langle (0,0,0,0,1,w^{12}) \rangle_{{\mathbb F}_{16}}, && L_8 = \langle (1,w^7,w^8,w^5,w^{14},w^{12}) \rangle_{{\mathbb F}_{16}}, \\ & L_8^{\Psi} = \langle (1,w^{13},w^2,w^5,w^{11},w^3) \rangle_{{\mathbb F}_{16}}, && L_9 = \langle (1,w^2,w^8,w^4,w^{10},w^9) \rangle_{{\mathbb F}_{16}}, \\ & L_9^{\Psi} = \langle (1,w^8,w^2,w,w^{10},w^6) \rangle_{{\mathbb F}_{16}}, && L_{10} = \langle (1,w^{11},w^{11},w^{13},w,w^9) \rangle_{{\mathbb F}_{16}}, \\ & L_{10}^{\Psi} = \langle (1,w^{14},w^{14},w^7,w^4,w^6) \rangle_{{\mathbb F}_{16}}, && L_{11} = \langle (1,w,w^2,w^6,w^3,w^9) \rangle_{{\mathbb F}_{16}}, \\ & L_{11}^{\Psi} = \langle (1,w^4,w^8,w^9,w^12,w^6)\rangle_{{\mathbb F}_{16}}. &&\end{align*}

Here $\Psi$ is the involutory collineation of $\mathrm{PG}(5, 4^2)$ fixing pointwise $\mathrm{PG}(5, 4)$ . The projectivity induced by the matrix (acting on the left)

\begin{align*}\begin{pmatrix} 1\;\;\;\;\; & e\;\;\;\;\; & 0\;\;\;\;\; & 1\;\;\;\;\; & 1\;\;\;\;\; & e^2 \\ e\;\;\;\;\; & 1\;\;\;\;\; & 1\;\;\;\;\; & 1\;\;\;\;\; & e^2\;\;\;\;\; & 0 \\ e^2\;\;\;\;\; & 1\;\;\;\;\; & e\;\;\;\;\; & 0\;\;\;\;\; & 1\;\;\;\;\; & 1 \\ 1\;\;\;\;\; & e^2\;\;\;\;\; & 1\;\;\;\;\; & e\;\;\;\;\; & 0\;\;\;\;\; & 1 \\ e\;\;\;\;\; & e^2\;\;\;\;\; & 1\;\;\;\;\; & e^2\;\;\;\;\; & e\;\;\;\;\; & e^2 \\ e^2\;\;\;\;\; & e\;\;\;\;\; & 0\;\;\;\;\; & e\;\;\;\;\; & e\;\;\;\;\; & 1\end{pmatrix}\end{align*}

maps ${\mathcal N}$ to the normal rational curve ${\mathcal N}_{6, 4^2}$ consisting of the points

\begin{align*} \{\langle (1,t,t^2,t^3,t^4,t^5) \rangle_{{\mathbb F}_{16}} \mid t \in {\mathbb F}_{16}\} \cup \{\langle (0,0,0,0,0,1)\rangle_{{\mathbb F}_{16}}\}.\end{align*}

Hence ${\mathcal N}$ is a normal rational curve of $\mathrm{PG}(5, 4^2)$ . Moreover, the line $\ell_i$ is the intersection of the line $\langle L_i, L_i^{\Psi}\rangle_{{\mathbb F}_{16}}$ with $\mathrm{PG}(5, 4)$ , if $i = 6, \dots, 11$ ; whereas the line $\ell_i$ is the intersection of the line tangent to ${\mathcal N}$ at the point $L_i$ with $\mathrm{PG}(5, 4)$ in the case when $i = 1,2,3,4,5$ .