Absolute points of correlations of $PG(3,q^n)$

Abstract

The sets of the absolute points of (possibly degenerate) polarities of a projective space are well known. The sets of the absolute points of (possibly degenerate) correlations, different from polarities, of ${{\mathrm{PG}}}(2,q^n)$, have been completely determined by B.C. Kestenband in 11 papers from 1990 to 2014, for non-degenerate correlations and by D’haeseleer and Durante (Electron J Combin 27(2):2–32, 2020) for degenerate correlations. In this paper, we completely determine the sets of the absolute points of degenerate correlations, different from degenerate polarities, of a projective space ${{\mathrm{PG}}}(3,q^n)$. As an application we show that, for q even, some of these sets are related to the Segre’s $(2^h+1)$-arc of ${{\mathrm{PG}}}(3,2^n)$ and to the Lüneburg spread of ${{\mathrm{PG}}}(3,2^{2h+1})$.

1 Introduction and preliminary results

1.1 Sesquilinear forms and correlations

Let V and W be two ${\mathbf {F}}$-vector spaces, where ${\mathbf {F}}$ is a field. A map $f:V\longrightarrow W$ is called semilinear or $\sigma$-linear if there exists an automorphism $\sigma$ of ${\mathbf {F}}$ such that

$$\begin{aligned} f(v+v') = f(v)+f(v') \, \, \, \, \text{ and } \, \, \, \, f(a v) = a^\sigma f(v) \end{aligned}$$

for all vectors $v\in V$ and all scalars $a\in {\mathbf {F}}$.

If $\sigma$ is the identity map, then f is a usual linear map.

Let V be an ${\mathbf {F}}$-vector space with finite dimension d. A map

$$\begin{aligned} \langle ~,~ \rangle : (v,v') \in V\times V \longrightarrow \langle v , v' \rangle \in {\mathbf {F}}\end{aligned}$$

is a sesquilinear form or a semibilinear form on V if it is a linear map on the first argument and it is a $\sigma$-linear map on the second argument, that is:

$$\begin{aligned} \langle v+v', v'' \rangle= & {} \langle v , v'' \rangle + \langle v', v''\rangle ,\\ \langle v, v'+v'' \rangle= & {} \langle v , v' \rangle + \langle v, v''\rangle ,\\ \langle a v, v' \rangle= & {} a \langle v, v' \rangle , \, \, \, \, \langle v, a v' \rangle = a^\sigma \langle v, v' \rangle , \end{aligned}$$

for all $v,v',v''\in V$, $a\in {\mathbf {F}}$ and $\sigma$ an automorphism of ${\mathbf {F}}$. If $\sigma$ is the identity map, then $\langle ~,~ \rangle$ is an usual bilinear form.

If ${{{\mathcal {B}}}} = (e_1,e_2,\ldots , e_{d+1})$ is an ordered basis of V, then for $x,y \in V$ we have $\langle x,y \rangle = X^t A Y^\sigma$, where $A=(\langle e_i, e_j \rangle )$ is the associated matrix to the sesquilinear form with respect to the ordered basis ${{{\mathcal {B}}}}$, X and Y are the columns of the coordinates of x, y w.r.t. ${{{\mathcal {B}}}}$. The term sequi comes from the Latin, and it means one and a half. For every subspace S of V, put

$$\begin{aligned} S^{\top }&:= \{y \in V: \langle x, y \rangle =0 \,\,\, \forall x \in S \},\\ S^\perp&:= \{y \in V: \langle y, x \rangle =0 \,\,\, \forall x \in S \}. \end{aligned}$$

Both $S^{\top }$ and $S^\perp$ are subspaces of V. The subspaces $V^{\top }$ and $V^\perp$ are called the right and the left radical of $\langle ~,~ \rangle$ and will be also denoted by $Rad_r(V)$ and $Rad_l(V)$, respectively.

Proposition 1.1

The right and the left radicals of a sesquilinear form of a vector space V have the same dimension.

Proof

Indeed, ${\mathrm{dim}}V^{\top } = d+1 - \text{ rk }(B)$, where $B=(a_{ij}^{\sigma ^{-1}})$ and ${\mathrm{dim}}V^\perp = d+1 -\text{ rk }(A_t)$, where $A_t$ is the transposed of the matrix A. The assertion follows from rk$(B)=\text{ rk }(A)=\text{ rk }(A_t)$. $\square$

A non-degenerate sesquilinear form $\langle ~,~ \rangle$ has ${V}^\perp ={V}^{\top }=\{0\}.$

Definition 1

A $\sigma$-sesquilinear form is reflexive if $\forall \; u,v \in V$ it is:

$$\begin{aligned} \langle u, v \rangle =0 \iff \langle v, u \rangle =0. \end{aligned}$$

Definition 2

Let V be an ${\mathbf {F}}$-vector space of dimension greater than two. A bijection $g:{{\mathrm{PG}}}(V) \longrightarrow {{\mathrm{PG}}}(V)$ is a collineation if g, together with $g^{-1}$, maps lines into lines. If V has dimension two, then a collineation is a map $\langle v\rangle \in {{\mathrm{PG}}}(V) \longrightarrow \langle f(v)\rangle \in {{\mathrm{PG}}}(V)$, induced by a bijective semilinear map $f:V\longrightarrow V$.

Theorem 1.2

(Fundamental Theorem) Let V be an ${\mathbf {F}}$-vector space. Every collineation of ${{\mathrm{PG}}}(V)$ is induced by a bijective semilinear map $f: V \longrightarrow V$.

In the sequel if S is a vector subspace of V, we will denote with the same symbol S the associated projective subspace of ${{\mathrm{PG}}}(V)$.

Definition 3

Let $f:V\longrightarrow V$ be a semilinear map, with $Ker f \ne \{0\}$. The map

$$\begin{aligned} \langle v\rangle \in {{\mathrm{PG}}}(V) {\setminus } Ker f \longrightarrow \langle f(v)\rangle \in {{\mathrm{PG}}}(V), \end{aligned}$$

will be called a degenerate collineation of ${{\mathrm{PG}}}(V)$.

Definition 4

A (degenerate) correlation or (degenerate) duality of ${{\mathrm{PG}}}(d,{\mathbf {F}})$ is a (degenerate) collineation between ${{\mathrm{PG}}}(d,{\mathbf {F}})$ and its dual space ${{\mathrm{PG}}}(d,{\mathbf {F}})^*$.

Remark 1.3

A correlation of ${{\mathrm{PG}}}(d,{\mathbf {F}})$ can be seen as a bijective map of ${{\mathrm{PG}}}(d,{\mathbf {F}})$ that maps k-dimensional subspaces into $(d-1-k)$-dimensional subspaces reversing inclusion and preserving incidence.

A correlation of ${{\mathrm{PG}}}(d,{\mathbf {F}})$ applied twice gives a collineation of ${{\mathrm{PG}}}(d,{\mathbf {F}})$.

Theorem 1.4

Any (possibly degenerate) correlation of ${{\mathrm{PG}}}(d,{\mathbf {F}})$, $d>1$, is induced by a $\sigma$-sesquilinear form of the underlying vector space ${\mathbf {F}}^{d+1}$. Conversely, every $\sigma$-sesquilinear form of ${\mathbf {F}}^{d+1}$ induces two (possibly degenerate) correlations of ${{\mathrm{PG}}}(d,{\mathbf {F}})$. The two correlations agree if and only if the form $\langle ~,~ \rangle$ is reflexive.

Proof

A (possibly degenerate) duality of ${{\mathrm{PG}}}(d,{\mathbf {F}})$, $d>1$, is induced by a $\sigma$-linear transformation f of ${\mathbf {F}}^{d+1}$ into its dual, since it is a (possibly degenerate) collineation. Define a map $\langle ~ , ~ \rangle : {\mathbf {F}}^{d+1}\times {\mathbf {F}}^{d+1} \longrightarrow {\mathbf {F}}$ in the following way:

$$\begin{aligned} \langle u, v \rangle = f(v)(u), \end{aligned}$$

that is the result of applying the element f(v) of $({\mathbf {F}}^{d+1})^*$ to u. It follows that $\langle ~ , ~ \rangle$ is a $\sigma$-sesquilinear form. Indeed, it is easy to see that $\langle ~ , ~ \rangle$ is linear on the first argument and since

$$\begin{aligned} \langle u, v_1+v_1 \rangle = f(v_1+v_2)(u)=f(v_1)(u)+f(v_2)(u) =\langle u, v_1\rangle + \langle u,v_2\rangle , \end{aligned}$$

and

$$\begin{aligned} \langle u, a v \rangle = f(av)(u) = a^\sigma f(v)(u) = a^\sigma \langle u, v \rangle , \end{aligned}$$

it is semilinear on the second argument. Moreover, $\langle ~ , ~ \rangle$ is non-degenerate if and only if f is a bijection.

Conversely, any $\sigma$-sesquilinear form on $V={\mathbf {F}}^{d+1}$ induces two (possibly degenerate) correlations of ${{\mathrm{PG}}}(d,{\mathbf {F}})$ given by

$$\begin{aligned} \perp : P=\langle u\rangle \in {{\mathrm{PG}}}(d,{\mathbf {F}}) {\setminus } Rad_l(V) \mapsto P^{\perp } =\{\langle v\rangle \, | \, \langle u, v \rangle =0 \} \in {{\mathrm{PG}}}(d,{\mathbf {F}})^* \end{aligned}$$

and

$$\begin{aligned} \top : P=\langle v\rangle \in {{\mathrm{PG}}}(d,{\mathbf {F}}) {\setminus } Rad_r(V) \mapsto P^{\top } =\{\langle u\rangle \, | \, \langle u, v \rangle =0 \} \in {{\mathrm{PG}}}(d,{\mathbf {F}})^*. \end{aligned}$$

$\square$

Remark 1.5

A (possibly degenerate) correlation induced by a $\sigma$-sesquilinear form will be also called a $\sigma$-correlation. Sometimes we will call linear a (degenerate) correlation whose associated form is a bilinear form.

1.2 Reflexive sesquilinear forms and polarities

Definition 5

A (degenerate) polarity is a (degenerate) correlation whose square is the identity.

If $\perp$ is a (possibly degenerate) polarity, then for every pair of points P and R the following holds:

$$\begin{aligned} P \in R^\perp \iff R\in P^\perp . \end{aligned}$$

Proposition 1.6

A (degenerate) correlation is a (degenerate) polarity if and only if the induced sesquilinear form is reflexive.

Proof

Let $\langle ~ , ~ \rangle$ be a reflexive $\sigma$-sesquilinear form of ${{\mathbb {F}}}_q^{d+1}$. If $u \in \langle v \rangle ^{\perp }$, then $v \in \langle u \rangle ^{\perp }$. Hence, the map $\langle u\rangle \mapsto \langle u\rangle ^\perp$ defines a (possibly degenerate) polarity. Conversely, given a (possibly degenerate) polarity $\perp$, if $v\in \langle u\rangle ^\perp$, then $u\in \langle v\rangle ^{\perp }$. So, the induced sesquilinear form is reflexive. $\square$

The non-degenerate, reflexive $\sigma$-sesquilinear forms of a $(d+1)$-dimensional ${\mathbf {F}}$-vector space V have been classified (for a proof see, e.g., Theorem 3.6 in [1] or Theorem 6.3 and Proposition 6.4 in [5]).

Theorem 1.7

Let $\langle ~ , ~ \rangle$ be a non-degenerate, reflexive $\sigma$-sesquilinear form of a $(d+1)$-dimensional ${\mathbf {F}}$-vector space. Then, up to a scalar factor, the form $\langle ~ , ~ \rangle$ is one of the following:

• (i) a symmetric form, i.e.,

  $$\begin{aligned} \forall \, \, u,v \in V \, \, \langle u, v \rangle = \langle v, u \rangle \,\,\, (\text{ char }({\mathbf {F}}) =2 \implies \exists \, v \in V: \langle v , v \rangle \ne 0), \end{aligned}$$

• (ii) an alternating form, i.e.,

  $$\begin{aligned} \forall \, \, v \in V \, \, \langle v ,v \rangle =0 \,\,\,\,\, (d \text{ is } \text{ necessarily } \text{ odd), } \end{aligned}$$

• (iii) a Hermitian form, i.e.,

  $$\begin{aligned} \forall \,\, u,v \in V \, \, \langle u,v\rangle = \langle v, u \rangle ^\sigma \,\,\, (\sigma ^2 =1, \sigma \ne 1). \end{aligned}$$

Let V be a $(d+1)$-dimensional ${\mathbf {F}}$-vector space and consider the associated projective geometry ${{\mathrm{PG}}}(d,{\mathbf {F}})$. If V is equipped with a sesquilinear from $\langle ~,~ \rangle$, we may consider in ${{\mathrm{PG}}}(d,{\mathbf {F}})$ the set $\Gamma$ of absolute points of the associated correlations, that is the points X such that $X\in {X}^\perp$ (or equivalently $X\in {X}^{\top })$. If A is the associated matrix to the $\sigma$-sesquilinear form $\langle ~,~ \rangle$ w.r.t. an ordered basis of V, then the set $\Gamma$ has equation $X^t A X^\sigma =0$.

From the previous theorem, the polarities of ${{\mathrm{PG}}}(d,{\mathbf {F}})$ are in one to one correspondence with non-degenerate, reflexive $\sigma$-sesquilinear forms of ${{\mathbf {F}}}^{d+1}$. Hence, to every polarity of ${{\mathrm{PG}}}(d,{\mathbf {F}})$ there is an associated pair $(A,\sigma )$, with A a non-singular matrix of order $d+1$ and $\sigma$ an automorphism of ${\mathbf {F}}$. In what follows, we will identify a polarity with a pair $(A,\sigma$). From the last theorem of the previous section, we have the following:

Theorem 1.8

Let $(A,\sigma )$ be a polarity of ${{\mathrm{PG}}}(d,q)$, one of the following holds:

• (i) $\sigma = 1$, A is a symmetric matrix. The polarity is called an orthogonal polarity. If q is even, there is a non-absolute point.

• (ii) $\sigma = 1$, A is a skew-symmetric matrix, d is odd. Every point is an absolute point and the polarity is called a symplectic polarity.

• (iii) $\sigma ^2= 1,$ so $\sigma : x \mapsto x^{\sqrt{q}},$ q is a square, A is a Hermitian matrix. The polarity is called a Hermitian or unitary polarity.

Recall that a square matrix A is either symmetric if $A=A_t$, or skew-symmetric if $A= -A_t$ or Hermitian if $A=A_t^{\sigma }, \sigma ^2=1, \sigma \ne 1$. Each of the above polarities of ${{\mathrm{PG}}}(d,q)$ determines a set $\Gamma :X^t A X^\sigma =0$, as set of its absolute points. The three types of polarities give rise to the following, well known, subsets of ${{\mathrm{PG}}}(d,q)$:

Definition 6

If $(A,\sigma )$ is a polarity of ${{\mathrm{PG}}}(d,q)$, then one of the following holds:

• (i) $\Gamma$ is called a quadric of ${{\mathrm{PG}}}(d,q)$ (q odd, orthogonal polarity). $\Gamma$ is a hyperplane of ${{\mathrm{PG}}}(d,q)$ (q even, orthogonal polarity).

• (ii) $\Gamma$ is the full pointset of ${{\mathrm{PG}}}(d,q)$ (d odd, symplectic polarity) and the geometry determined is a symplectic polar space.

• (iii) $\Gamma$ is a Hermitian variety of ${{\mathrm{PG}}}(d,q)$ (q a square, $\sigma ^2=1, \sigma \ne 1$, unitary polarity).

Remark 1.9

If q is even, $\sigma =1$ and $(A,\sigma )$ is an orthogonal polarity, so A is a symmetric matrix, then the set $\Gamma$ of its absolute points is a hyperplane. Note that in many books (but not in the book of P. Dembowski [10]) this kind of polarity is called a pseudo polarity.

In all the cases of the previous definition, the set $\Gamma$ is often called also non-degenerate since the associated $\sigma$-sesquilinear form is non-degenerate. All the above sets have been classified, and for each of them it is possible to give a canonical equation (see, e.g., chapters 22 and 23 in [16]).

Theorem 1.10

If $(A,\sigma )$ is a polarity of ${{\mathrm{PG}}}(d,q)$, then let $\Gamma :X^t A X^\sigma =0$ be the set of its absolute points. The following hold:

• (i) If $\Gamma$ is a quadric, so q is odd, then we have:

  • (1) If d is even, then

    $$\begin{aligned} \Gamma ={\mathcal {Q}}(d,q): x_1x_2+\cdots +x_{d-1}x_d+ x_{d+1}^2=0, \quad (\text{ parabolic } \text{ quadric}). \end{aligned}$$

  • (2) If d is odd, then either

    $$\begin{aligned} \Gamma ={\mathcal {Q}}^-(d,q):x_1x_2+\cdots +\alpha x_{d}^2+\beta x_{d} x_{d+1} +\gamma x_{d+1}^2=0, with \end{aligned}$$

    $\alpha x_{d}^2+\beta x_{d} x_{d+1}+\gamma x_{d+1}^2$ irreducible polynomial over ${{\mathbb {F}}}_q$ $(\text{ elliptic } \text{ quadric})$ or

    $$\begin{aligned} \Gamma ={\mathcal {Q}}^+(d,q):x_1x_2+\cdots +x_{d}x_{d+1}=0 \quad (\text{ hyperbolic } \text{ quadric}). \end{aligned}$$

  If $\Gamma$ is a hyperplane, so q is even, then $\Gamma :x_1=0$.

• (ii) If $\Gamma$ is a symplectic polar space, then $\Gamma$ is the full pointset of ${{\mathrm{PG}}}(d,q), d$ odd. The geometry determined will be denoted by W(d, q). A canonical form for the associated bilinear form is

  $$\begin{aligned} \langle x, y \rangle = x_1y_2-x_2y_1+x_3y_4-x_4y_3+\cdots +x_{d}y_{d+1}-x_{d+1} y_{d}. \end{aligned}$$

• (iii) If $\Gamma$ is a Hermitian variety, then

  $$\begin{aligned} \Gamma ={{{\mathcal {H}}}}(d,q):x_1^{\sqrt{q}+1}+x_2^{\sqrt{q}+1}+\cdots +x_{d+1}^{\sqrt{q}+1}=0. \end{aligned}$$

Remark 1.11

Regarding degenerate, reflexive $\sigma$-sesquilinear forms of $V={{\mathbb {F}}}_q^{d+1}$, it is possible to prove that if dim $V^\perp = r+1$, $r\ge 0$, then the set $\Gamma$ of absolute points in ${{\mathrm{PG}}}(d,q)$ of the associated degenerate polarity is, in all the possible cases, a cone $\Gamma ({{{\mathcal {V}}}}_r, Q_{d-1-r})$ with vertex a subspace ${{{\mathcal {V}}}}_r$, of dimension r, projecting the set $Q_{d-1-r}$ of absolute points of a polarity in a subspace $S_{d-1-r}$, of dimension $d-1-r$, skew with ${{{\mathcal {V}}}}_r$. The set $Q_{d-1-r}$ can be either a quadric or a Hermitian variety (if q is a square) or the full pointset of ${{\mathrm{PG}}}(d-1-r,q)$ (a symplectic geometry). In these cases, we call the set of the absolute points either a degenerate quadric,or a degenerate Hermitian variety or a degenerate symplectic geometry, respectively. The knowledge of the set of the absolute points of a polarity of $S_{d-1-r}$ determines also the knowledge of the set of the absolute points of a degenerate polarity. If $Q_{d-1-r}$ is the full pointset of ${{\mathrm{PG}}}(d-1-r)$, then d and r must have the same parity and the cone $\Gamma ({{{\mathcal {V}}}}_r, {{\mathrm{PG}}}(d-1-r,q))$ is a quotient geometry ${{\mathrm{PG}}}(d,q)/{{{\mathcal {V}}}}_r$.

Sometimes we will call non-degenerate quadrics, non-degenerate symplectic geometries and non-degenerate Hermitian varieties, the set of the absolute points of a polarity. The following proposition characterizes these sets.

Proposition 1.12

Let $\Gamma$ be either a non-degenerate quadric or a non-degenerate symplectic polar space or a non-degenerate Hermitian variety of ${{\mathrm{PG}}}(d,q)$ and denote by $\perp$ the associated polarity. For any point Y of $\Gamma$, the set $Y^\perp$ is a hyperplane meeting $\Gamma$ in a cone $\Gamma (Y,Q_{d-2})$, where $Q_{d-2}$ is either a non-degenerate quadric or a non-degenerate symplectic polar space or a non-degenerate Hermitian variety, respectively and vice versa.

1.3 Sets of the absolute points of a linear correlation of ${{\mathrm{PG}}}{(d,{\mathbf {F}})}$

Let $\Gamma$ be the set of the absolute points of a linear correlation of ${{\mathrm{PG}}}(d,{\mathbf {F}})$ with equation $X^t A X=0$, $|A|\ne 0$. Assuming that $\Gamma$ is a proper subset of ${{\mathrm{PG}}}(d,{\mathbf {F}})$, the matrix A cannot be a skew-symmetric matrix and $\Gamma$ is a (possibly degenerate) quadric and vice versa. Note that the previous holds, independently of the characteristic of the field ${\mathbf {F}}$. If the characteristic of ${\mathbf {F}}$ is 2, then there is no relation between |A| and degeneracy or not of the quadric. Observe that, not assuming that A is a symmetric matrix, also for characteristic of ${\mathbf {F}}$ either odd or 0, this relation is lost and, a bit surprisingly, the following holds:

Proposition 1.13

If $\Gamma :X^t A X=0$ is a proper subset of points of ${{\mathrm{PG}}}(d,{\mathbf {F}})$, with $|A|=0$, then $\Gamma$ is a (possibly degenerate) quadric. Vice versa, if $\Gamma$ is a non-empty (possibly degenerate) quadric of ${{\mathrm{PG}}}(d,{\mathbf {F}})$, then there exists a matrix A, with $|A|=0$, s.t. $\Gamma$ is projectively equivalent to the set of points with equation $X^t A X=0$.

Moreover, since in the finite Desarguesian projective spaces all the quadrics are non-empty (see, e.g., [38]), we have the following:

Proposition 1.14

If $\Gamma : X^t A X=0$, with $|A|=0$, is a set of points of ${{\mathrm{PG}}}(d,q^n)$, then $\Gamma$ is either a degenerate symplectic geometry or a (possibly degenerate) quadric. Vice versa if $\Gamma$ is either a degenerate symplectic geometry or a (possibly degenerate) quadric of ${{\mathrm{PG}}}(d,q^n)$, then there is a matrix A with $|A|=0$ s.t. $\Gamma$ is projectively equivalent to the set of points with equation $X^t A X=0$.

1.4 $\pmb {{\mathbb {F}}_{{q}}}$-linear sets of ${{\mathrm{PG}}}({d,q^n})$

Definition 7

Let $\Omega ={{\mathrm{PG}}}(r-1,q^n), q=p^h$, p a prime. A set L is said an ${{\mathbb {F}}}_q$-linear set of $\Omega$ of rank t if it is defined by the nonzero vectors of an ${{\mathbb {F}}}_q$-vector subspace U of $V={\mathbf {F}}_{q^n}^{r}$ of dimension t, that is

$$\begin{aligned} L=L_U =\{\langle u\rangle _{q^n}: {u} \in U{\setminus } \{{0}\}\}. \end{aligned}$$

Let $\Lambda ={{\mathrm{PG}}}(W,{{\mathbb {F}}}_{q^n})$ be a subspace of $\Omega$ of dimension s, we say that $\Lambda$ has weight i with respect to $L_U$ if $\hbox {dim}_{{{\mathbb {F}}}_q} (W \cap U) =i$. An ${{\mathbb {F}}}_q$-linear set $L_U$ of $\Omega$ of rank t is said to be scattered if each point of $L_U$ has weight 1, with respect to $L_U$.

In [2] it is proved that if $L_U$ is a scattered ${{\mathbb {F}}}_q$ linear set of $\Omega$, then $t\le rn/2$. It is clear, by definition that $L_U$ is a scattered ${{\mathbb {F}}}_q$-linear set of rank t if and only if $|L_U|=q^{t-1}+q^{t-2}+\cdots +q+1$. If L is a scattered linear set of ${{\mathrm{PG}}}(r-1,q^n)$ of rank rn/2, it is called a maximum scattered linear set.

If $\hbox {dim}_{{{\mathbb {F}}}_q} U=$ $\hbox {dim}_{{{\mathbb {F}}}_{q^n}} V =r$ and $\langle U\rangle _{{{\mathbb {F}}}_{q^n}} =V$, then $L_U \cong {{\mathrm{PG}}}(U,{{\mathbb {F}}}_q)$ is a subgeometry of $\Omega$. In such a case, each point has weight 1, and hence $|L_U|=q^{r-1}+q^{r-2}+ \cdots +q+1$. Let $\Sigma ={{\mathrm{PG}}}(t,q)$ be a subgeometry of $\Sigma '={{\mathrm{PG}}}(t,q^n)$ and suppose that there exists a $(t-r)$-dimensional subspace $\Omega '$ of $\Sigma '$ disjoint from $\Sigma$. Let $\Omega ={{\mathrm{PG}}}(r-1,q^n)$ be an $(r-1)$-dimensional subspace of $\Sigma '$ disjoint from $\Omega '$, and let $\Gamma$ be the projection of $\Sigma$ from $\Omega '$ to $\Omega$, i.e., $\Gamma = \{ \langle \Omega ', x\rangle \cap \Omega : x \in \Sigma \}$. Let $p_{\Omega ',\Omega }$ be the map from $\Sigma {\setminus } \Omega '$ to $\Omega$ defined by $x \mapsto \langle \Omega ', x\rangle \cap \Omega$. We call $\Omega '$ the center and $\Omega$ the axis of $p_{\Omega ',\Omega }$. In [31] the following characterization of ${{\mathbb {F}}}_q$-linear sets is given:

Theorem 1.15

If L is a projection of ${{\mathrm{PG}}}(t,q)$ into $\Omega ={{\mathrm{PG}}}(r-1,q^n)$, then L is an ${{\mathbb {F}}}_q$-linear set of $\Omega$ of rank $t+1$ and $\langle L \rangle =\Omega$. Conversely, if L is an ${{\mathbb {F}}}_q$-linear set of $\Omega$ of rank $t+1$ and $\langle L \rangle =\Omega$, then either L is a subgeometry of $\Omega$ or there are a $(t-r)$-dimensional subspace $\Omega '$ of $\Sigma '={{\mathrm{PG}}}(t,q^n)$ disjoint from $\Omega$ and a subgeometry $\Sigma$ of $\Sigma '$ disjoint from $\Omega '$ such that $L=p_{\Omega ',\Omega }(\Sigma )$.

A family of maximum scattered linear sets to which a geometric structure, called pseudoregulus, can be associated has been defined in [30].

Definition 8

Let $L = L_U$ be a scattered ${{\mathbb {F}}}_q$-linear set of $\Gamma = {{\mathrm{PG}}}(2n-1,q^t)$ of rank $tn, t,n \ge 2.$ We say that L is of pseudoregulus type if:

• (i) there exist $m = \frac{q^{nt}-1}{q^t-1}$ pairwise disjoint lines of $\Gamma$, say $s_1,s_2,\ldots ,s_m$, such that $w_L(s_i) = t, i.e., |L \cap s_i| = q^{t-1} + q^{t-2} +\cdots + q + 1 \,\,\, \forall \, i =1,\ldots ,m$,

• (ii) there exist exactly two $(n-1)$-dimensional subspaces $T_1$ and $T_2$ of $\Gamma$ disjoint from L such that $T_j \cap s_i \ne \emptyset \,\,\,$ $\forall \, i = 1,\ldots ,m$ and $j = 1,2$.

We call the set of lines ${{{\mathcal {P}}}}_L = \{s_i: i = 1,\ldots ,m\}$ the ${{\mathbb {F}}}_q$-pseudoregulus (or simply the pseudoregulus) of $\Gamma$ associated with L and we refer to $T_1$ and $T_2$ as transversal spaces of ${{{\mathcal {P}}}}_L$ (or transversal spaces of L). When $t=n=2$, these objects already appeared first in [15], where the term pseudoregulus was introduced for the first time. See also [11].

In [12] and in [30] the following class of maximum scattered ${{\mathbb {F}}}_q$-linear sets of the projective line $\Gamma ={{\mathrm{PG}}}(1,q^t)$ with a structure resembling that of an ${{\mathbb {F}}}_q$-linear set of ${{\mathrm{PG}}}(2n-1,q^t), n,t \ge 2,$ of pseudoregulus type has been studied. Let $P_1 =\langle w\rangle _{q^t}$ and $P_2 = \langle v\rangle _{q^t}$ be two distinct points of $\Gamma$ and let $\tau$ be an automorphism of ${\mathbb {F}}_{q^t}$ such that $Fix(\tau ) = {{\mathbb {F}}}_q$. For each $\rho \in {\mathbb {F}}_{q^t}^*$, the set $W_{\rho ,\tau } = \{\lambda w + \rho \lambda ^\tau v :\lambda \in {\mathbb {F}}_{q^t}\}$, is an ${{\mathbb {F}}}_q$-vector subspace of $V={\mathbb {F}}_{q^t}^2$ of dimension t and $L_{\rho ,\tau } = L_{W_{\rho ,\tau }}$ is a scattered ${{\mathbb {F}}}_q$-linear set of $\Gamma$.

Definition 9

In [30] the linear sets $L_{\rho ,\tau }$ have been called of pseudoregulus type and the points $P_1$ and $P_2$ the transversal points of $L_{\rho ,\tau }$. If $L_{\rho ,\tau }\cap L_{\rho ',\tau } \ne \emptyset$, then $L_{\rho ,\tau }= L_{\rho ',\tau }$.

Remark 1.16

(See [12, 30]) Note that $L_{\rho ,\tau }= L_{\rho ',\tau }$ if and only if $N(\rho ) = N(\rho ')$ (where N denotes the Norm of ${{\mathbb {F}}}_{q^t}$ over ${{\mathbb {F}}}_q)$; so $P_1, P_2$ and the automorphism $\tau$ define a set of $q-1$ mutually disjoint maximum scattered linear sets of pseudoregulus type admitting the same transversal points. Such maximal scattered linear sets, together with $P_1$ and $P_2$, cover the pointset of ${{\mathrm{PG}}}(1,q^t)$. All the ${{\mathbb {F}}}_q$-linear sets of pseudoregulus type in $\Gamma = {{\mathrm{PG}}}(1,q^t), t\ge 2$, are equivalent to the linear set $L_{1,\sigma _1}$ under the action of the collineation group ${\mathrm {P}}\Gamma {}{\mathrm {L}}(2,q^t)$.

Remark 1.17

We observe that the pseudoregulus of ${{\mathrm{PG}}}(1,q^n)$ is the same as the Norm surface (or sphere) of R.H. Bruck, introduced and studied, in the seventies, by R.H. Bruck (see [3, 4]) in the setting of circle geometries.

For more on ${{\mathbb {F}}}_q$-linear sets in finite projective spaces, see, e.g., [28, 33].

2 The $\sigma$-quadrics of ${{\mathrm{PG}}}(d,q^n)$

In this section, we will introduce $\sigma$-quadrics of ${{\mathrm{PG}}}(d,q^n)$ and we will determine their intersection with subspaces.

Definition 10

A $\sigma$-quadric of ${{\mathrm{PG}}}(d,q^n)$ is the set of the absolute points of a (possibly degenerate) $\sigma$-correlation, $\sigma \ne 1$, of ${{\mathrm{PG}}}(d,q^n)$. A $\sigma$-quadric of ${{\mathrm{PG}}}(2,q^n)$ will be called a $\sigma$-conic.

Remark 2.1

Let $\Gamma :X^t A X^\sigma =0$ be the set of the absolute points of a (possibly degenerate) $\sigma$-correlation of ${{\mathrm{PG}}}(d,q^2)$ and let $\sigma :x\mapsto x^q$. In this case, the set $\Gamma$ is a (possibly degenerate) Hermitian variety of ${{\mathrm{PG}}}(d,q^2)$ if A is a Hermitian matrix. We have included Hermitian varieties in the definition of a $\sigma$-quadric in order to have no exceptions everywhere. Indeed, we will see that Hermitian varieties appear as intersection of $\sigma$-quadrics of ${{\mathrm{PG}}}(d,q^2)$ with subspaces.

Proposition 2.2

Let $\Gamma :X^t A X^\sigma =0$ be a $\sigma$-quadric of ${{\mathrm{PG}}}(d,q^n)$. Every subspace S intersects $\Gamma$ either in a $\sigma$-quadric of S or it is contained in $\Gamma$.

Proof

Let S be an h-dimensional subspace of ${{\mathrm{PG}}}(d,q^n)$. We may assume, w.l.o.g. that $S:x_{h+2}=0, \ldots , x_{d+1}=0$. Let $A'$ be the submatrix of A obtained by deleting the last $d-h$ rows and columns of A. If $A'=0$, then $S\subset \Gamma$. If $A'\ne 0$, then $S\cap \Gamma$ is a $\sigma$-quadric of S. $\square$

For the remaining part of this chapter, we can assume $\sigma \ne 1$. Let $V={{\mathbb {F}}}_{q^n}^{d+1}$, let $\langle ~,~ \rangle$ be a degenerate $\sigma$-sesquilinear form with associated (degenerate) correlations $\perp$, $\top$ and let $\Gamma :X^t A X^\sigma =0$ be the associated $\sigma$-quadric. We will denote by $L=V^{\perp }$ and $R=V^{\top }$, the left and right radicals of $\langle ~,~ \rangle$ respectively, seen as subspaces of ${{\mathrm{PG}}}(d,q^n)$ that will be called the vertices of $\Gamma$. Before giving the definition of a non-degenerate $\sigma$-quadric, we prove the following:

Proposition 2.3

Let $\Gamma :X^t A X^\sigma =0$ be a $\sigma$-quadric of ${{\mathrm{PG}}}(d,q^n)$.

• For every point $Y\in \Gamma {\setminus } R$, the hyperplane $Y^{\top }$ is union of lines through Y either contained or 1-secant or 2-secant to $\Gamma$.

• For every point $Y\in \Gamma {\setminus } L$, the hyperplane $Y^{\perp }$ is union of lines through Y either contained or 1-secant or 2-secant to $\Gamma$.

Proof

Let $Y\in \Gamma {\setminus } R$ and let Z be a point of $Y^{\top }$. The line YZ has equations: $X=\lambda Y + \mu Z, (\lambda ,\mu ) \in {{\mathrm{PG}}}(1,q^n)$; hence, $Y^{\top }\cap \Gamma$ is determined by the solutions in $(\lambda ,\mu )$ of the following equation:

$$\begin{aligned} Y^t A Y^\sigma \lambda ^{\sigma +1} + Y^t A Z^\sigma \lambda \mu ^\sigma + Z^t A Y^\sigma \lambda ^\sigma \mu + Z^t A Z^\sigma \mu ^{\sigma +1}=0. \end{aligned}$$

(1)

In the previous equation, it is $Y^t A Y^\sigma =0$, since $Y\in \Gamma$ and $Z^t A Y^\sigma =0$, since $Z\in Y^{\top }$. Hence, Equation (1) becomes $Y^t A Z^\sigma \lambda + Z^t A Z^\sigma \mu =0$ and four different cases occur

• If $Y^t A Z^\sigma = Z^t A Z^\sigma =0$, then the line YZ is contained in $\Gamma$;

• If $Y^t A Z^\sigma =0, Z^t A Z^\sigma \ne 0$, then the line YZ intersects $\Gamma$ exactly at the point Y;

• If $Y^t A Z^\sigma \ne 0, Z^t A Z^\sigma =0$, then the line YZ intersects $\Gamma$ exactly at the point Y;

• If $Y^t A Z^\sigma \ne 0$, $Z^t A Z^\sigma \ne 0$, then the line YZ intersects $\Gamma$ at Y and at another point.

The second part of the statement follows in a similar way. $\square$

Corollary 2.4

Let $\Gamma :X^t A X^\sigma =0$ be a $\sigma$-quadric of ${{\mathrm{PG}}}(d,q^n)$ and let $L=V^\perp$, $R=V^{\top }$.

• For every point $Y\in L$, the set $Y^{\top }\cap \Gamma$ is union of lines through Y.

• For every point $Y\in R$, the set $Y^{\perp }\cap \Gamma$ is union of lines through Y.

Remark 2.5

Let $\Gamma :X^t A X^\sigma =0$ be a $\sigma$-quadric of ${{\mathrm{PG}}}(d,q^n)$ and let $L=V^\perp$ and $R=V^{\top }$ be its vertices. For every point $Y\in \Gamma {\setminus } R$, $Y^{\top }$ is the tangent hyperplane at Y to the hypersurface with equation $X^t A X^\sigma =0$ and for every point $Y\in \Gamma {\setminus } L$, $Y^{\perp }$ is the tangent hyperplane at Y to the hypersurface with equation $X^t A_t^{\sigma ^{-1}} X^{\sigma ^{-1}}=0$. Both these hypersurfaces have as set of ${{\mathbb {F}}}_{q^n}$-rational points the set $\Gamma$.

Inspired by the characterization of non-degenerate quadrics, Hermitian varieties and symplectic polar spaces of ${{\mathrm{PG}}}(d,q^n)$ (see Proposition 1.12), we define a non-degenerate $\sigma$-quadric of ${{\mathrm{PG}}}(d,q^n)$ by induction on the dimension d of the projective space.

Definition 11

Let $\Gamma$ be a $\sigma$-quadric of ${{\mathrm{PG}}}(d,q^n)$, denote by L and R the left and right radicals of the associated sesquilinear form.

• (i) $\Gamma$ is a non-degenerate $\sigma$-quadric of ${{\mathrm{PG}}}(1,q^n)$ if $|\Gamma |\in \{0, 2, q+1\}$.

• (ii) $\Gamma$ is a non-degenerate $\sigma$-conic of ${{\mathrm{PG}}}(2,q^n)$, if the following hold:

  • $L\cap R = \emptyset$,

  • the tangent line $L^{\top }$ to $\Gamma$ at L intersects $\Gamma$ exactly at the point L.

• (iii) $\Gamma$ is non-degenerate $\sigma$-quadric of ${{\mathrm{PG}}}(d,q^n), d\ge 3$, if the following hold:

  • $L\cap R = \emptyset$,

  • $\forall \; Y\in L$, the set $Y^{\top } \cap \Gamma$ is a cone $\Gamma (Y,Q)$ with vertex Y and base a non-degenerate $\sigma$-quadric Q in a $(d-2)$-dimensional subspace S not through Y.

2.1 The $\sigma$-quadrics of ${{\mathrm{PG}}}(1,q^n)$ and of ${{\mathrm{PG}}}(2,q^n)$

In this subsection, we recall what is known for the $\sigma$-quadrics of ${{\mathrm{PG}}}(1,q^n)$ and of ${{\mathrm{PG}}}(2,q^n)$ with $Fix(\sigma )={{\mathbb {F}}}_q$ (see [9, 14]).

In what follows, we will always assume that $\sigma \ne 1$ since if $\sigma =1$, the set $\Gamma$ is either a (possibly degenerate) quadric or a (possibly degenerate) symplectic geometry. Moreover, since $Fix(\sigma )={{\mathbb {F}}}_q$ we have that $\sigma :x\mapsto x^{q^m}$, $(m,n)=1$.

Proposition 2.6

A $\sigma$-quadric of ${{\mathrm{PG}}}(1,q^n)$ is one of the following:

• the empty set, a point, two distinct points,

• an ${{\mathbb {F}}}_q$-subline ${{\mathrm{PG}}}(1,q)$ of ${{\mathrm{PG}}}(1,q^n)$.

Obviously the following holds:

Proposition 2.7

Let $\Gamma$ and $\Gamma '$ be a $\sigma$-quadric and $\sigma '$-quadric of ${{\mathrm{PG}}}(1,q^n)$, respectively. The sets $\Gamma$ and $\Gamma '$ are ${\mathrm {P}}\Gamma {}{\mathrm {L}}$-equivalent if and only if $|\Gamma |=|\Gamma '|$.

The sets of the absolute points of $\sigma$-correlations of ${{\mathrm{PG}}}(2,q^n)$ have been completely determined by the huge work of B.C. Kestenband in 11 different papers from 1990 to 2014. There are lots of different classes of such sets. The interested reader can find all of them in 11 different papers from 1990 to 2014 covering 400 pages of mathematics (see [17,18,19,20,21,22,23,24,25,26,27]). In what follows, we will use the following:

Definition 12

A Kestenband $\sigma$-conic of ${{\mathrm{PG}}}(2,q^n)$ is the set of absolute points of a $\sigma$-correlation, $\sigma \ne 1$, of ${{\mathrm{PG}}}(2,q^n)$.

Proposition 2.8

A $\sigma$-conic of ${{\mathrm{PG}}}(2,q^n)$ is one of the following:

• a cone with vertex a point A projecting a $\sigma$-quadric of a line $\ell$ not through A.

• a degenerate $C_F^m$-set (i.e., the union of a line $\ell$ with a scattered linear set isomorphic to ${{\mathrm{PG}}}(n,q)$, meeting the line $\ell$ in a maximum scattered ${{\mathbb {F}}}_q$-linear set of pseudoregulus type),

• a $C_F^m$-set (i.e., the union of $q-1$ scattered ${{\mathbb {F}}}_q$-linear sets each of which isomorphic to ${{\mathrm{PG}}}(n-1,q)$ with two distinct points ).

• a Kestenband $\sigma$-conic

Proposition 2.9

Let $\Gamma$ be a $\sigma$-conic of ${{\mathrm{PG}}}(2,q^n)$ with two different vertices R and L. Associated to $\Gamma$ there is a $\sigma$-collineation between the pencils of lines with vertices R and L. Moreover, $\Gamma$ is also a $\sigma ^{-1}$ conic.

Proof

To the $\sigma$-conic $\Gamma$, there is associated a $\sigma$-collineation $\Phi$ between the pencils of lines with vertices R and L s.t. $\Gamma$ is the set of points of intersection of corresponding lines under $\Phi$ (resp. $\Phi '$) (see [9]). Moreover, the $\sigma$-conic $\Gamma$ with associated the $\sigma$-collineation $\Phi$ is also a $\sigma ^{-1}$ conic with associated the $\sigma ^{-1}$-collineation $\Phi ^{-1}$. $\square$

Proposition 2.10

Let $\Gamma$ be a $\sigma$-conic of ${{\mathrm{PG}}}(2,q^n)$ with two different vertices R, L and let $\Gamma '$ be a $\sigma '$-conic of ${{\mathrm{PG}}}(2,q^n)$ with two different vertices $R'$, $L'$. The sets $\Gamma$ and $\Gamma '$ are ${\mathrm {P}}\Gamma {}{\mathrm {L}}$-equivalent if and only if either $\sigma '=\sigma$ or $\sigma '=\sigma ^{-1}$.

Proof

Let $\sigma :x\mapsto x^{q^m}$ and let $\sigma ':x\mapsto x^{q^{m'}}$ with $(m,n)=(m',n)=1$. We may assume that $R'=R$ and $L'=L$ since the group ${\mathrm {P}}\Gamma {}{\mathrm {L}}(3,q^n)$ is two-transitive on points. Let $\Phi$ (resp. $\Phi '$) be the $\sigma$-collineation (resp. $\sigma '$-collineation) associated with $\Gamma$ (resp. $\Gamma '$). First, assume that $\Gamma$ is $C_F^m$-set and that $\Gamma '$ is a $C_F^{m'}$-set. Let f be a collineation of ${{\mathrm{PG}}}(2,q^n)$ mapping ${\Gamma }$ into ${\Gamma }'$. Since R and L are the unique points of both ${\Gamma }$ and ${\Gamma }'$ not incident with $(q+1)$-secant lines, it follows that f stabilizes the set $\{R,L\}$. First, assume that $f(R)=L$. For every line $\ell$ through the point R, we have that $f(\Phi (\ell ))=(\Phi ')^{-1}(f(\ell ))$. As $\Phi$ and $(\Phi ')^{-1}$ are collineations with accompanying automorphism $\sigma$ and $\sigma '^{-1}$, it follows that $\sigma '=\sigma ^{-1}$. Next suppose that $f(R)=R, f(L)=L$. For every line $\ell$ through R, we have that $f(\Phi (\ell ))=\Phi '(f(\ell ))$, and hence $\sigma =\sigma '$.

Next assume that $\Gamma$ is a degenerate $C_F^m$-set and that $\Gamma '$ is a degenerate $C_F^{m'}$-set and that $n\ge 3$. Let f be a collineation of ${{\mathrm{PG}}}(2,q^n)$ mapping ${\Gamma }$ into ${\Gamma }'$. Since the line RL is the unique line contained in both $\Gamma$ and $\Gamma '$, the collineation f stabilizes RL. The directions of the sets $\Gamma {\setminus } RL$ and $\Gamma '{\setminus } RL$ on the line RL are both ${{\mathbb {F}}}_q$-linear sets of pseudoregulus type with transversal points R and L. Since the transversal points of an ${{\mathbb {F}}}_q$-linear set of pseudoregulus type of ${{\mathrm{PG}}}(1,q^n)$, $n\ge 3$ are uniquely determined (see Proposition 4.3 in [30]), it follows that f stabilizes the set $\{R,L\}$. The assertion follows with the same arguments as in the previous case. $\square$

Finally, the following holds:

Proposition 2.11

Let $\Gamma$ be a $\sigma$-conic of ${{\mathrm{PG}}}(2,q^n)$ with vertices $R=L$, and let $\Gamma '$ be a $\sigma '$-conic of ${{\mathrm{PG}}}(2,q^n)$ with vertices $R'=L'$. The sets $\Gamma$ and $\Gamma '$ are ${\mathrm {P}}\Gamma {}{\mathrm {L}}$-equivalent if and only if $|\Gamma |=|\Gamma '|$.

Proof

We may assume that $R=R'$ since the group ${\mathrm {P}}\Gamma {}{\mathrm {L}}(3,q^n)$ is transitive on points. Obviously, if $\Gamma$ and $\Gamma '$ are ${\mathrm {P}}\Gamma {}{\mathrm {L}}$-equivalent, then $|\Gamma | = |\Gamma '|$. Vice versa if $|\Gamma | = |\Gamma '|$, then both $\Gamma$ and $\Gamma '$ are cones with vertex the point R projecting a $\sigma$-quadric on a line $\ell$ not through R. The assertion follows since two $\sigma$-quadrics of $\ell$ are ${\mathrm {P}}\Gamma {}{\mathrm {L}}$-equivalent if and only if have the same size (see [9]). $\square$

Remark 2.12

We can divide the $\sigma$-conics in two families:

• the degenerate $\sigma$-conics: the degenerate $C_F^m$-sets and the cones with vertex a point V projecting either the empty set or a point or two points or $q+1$ points on a line $\ell$ not through the point V;

• the non-degenerate $\sigma$-conics: the Kestenband $\sigma$-conics and the $C_F^m$-sets.

In the remaining part of this chapter, we will determine the canonical equations and some properties for the $\sigma$-quadrics of ${{\mathrm{PG}}}(3,q^n)$ associated to a degenerate correlation of ${{\mathrm{PG}}}(3,q^n)$. Let $\Gamma$ be such a $\sigma$-quadric with equation $X^t A X^\sigma =0$, A a singular matrix. We will consider three separate cases according to $rk(A) \in \{1,2,3\}$.

3 Seydewitz’s and Steiner’s projective generation of quadrics of ${{\mathrm{PG}}}(3,{\mathbf {F}})$

In this section, we recall the projective generation of F. Seydewitz and J. Steiner of quadrics of ${{\mathrm{PG}}}(3,{\mathbf {F}})$, ${\mathbf {F}}$ a field. We start with Seydewitz’s construction. In what follows, if P is a point of ${{\mathrm{PG}}}(3,{\mathbf {F}})$ we will denote with ${{{\mathcal {S}}}}_P$ the star of lines with center P and with ${{{\mathcal {S}}}}^*_P$ the star of planes with center P.

Theorem 3.1

(F. Seydewitz [36]) Let R and L be two distinct points of ${{\mathrm{PG}}}(3,{\mathbf {F}})$ and let $\Phi : {{{\mathcal {S}}}}_R \longrightarrow {{{\mathcal {S}}}}^*_L$ be a projectivity. The set $\Gamma$ of points of intersection of corresponding elements under $\Phi$ is one of the following:

• If $\Phi (RL)$ is a plane through the line RL, then $\Gamma$ is either a quadratic cone or a hyperbolic quadric ${\mathcal {Q}}^+(3,{\mathbf {F}})$,

• If $\Phi (RL)$ is a plane not through the line RL, then $\Gamma$ is a non-empty, non-degenerate quadric, i.e., either an elliptic quadric ${\mathcal {Q}}^-(3,{\mathbf {F}})$ or an hyperbolic quadric ${\mathcal {Q}}^+(3,{\mathbf {F}})$.

Next consider Steiner’s construction. In what follows, if s is a line of ${{\mathrm{PG}}}(3,{\mathbf {F}})$ we will denote by ${{{\mathcal {P}}}}_s$ the pencil of planes through s.

Theorem 3.2

(J. Steiner [37]) Let r and $\ell$ be two skew lines of ${{\mathrm{PG}}}(3,{\mathbf {F}})$. Let $\Phi : {{{\mathcal {P}}}}_r \longrightarrow {{{\mathcal {P}}}}_{\ell }$ be a projectivity. The set of points of intersection of corresponding planes under $\Phi$ is a hyperbolic quadric ${\mathcal {Q}}^+(3,{\mathbf {F}})$ of ${{\mathrm{PG}}}(3,{\mathbf {F}})$.

Proposition 3.3

In ${{\mathrm{PG}}}(2,{\mathbf {F}})$ the set of the absolute points of a linear (possibly degenerate) correlation satisfies an equation $X^t A X =0$, for some matrix A. Therefore, it is one of the followings:

• the empty set (e.g., ${\mathbf {F}}={\mathbb {R}}$)

• a point, a line, two lines,

• a non-degenerate conic,

• a degenerate symplectic geometry ${{{\mathcal {P}}}}_R$, for some point R.

Proof

It follows immediately from Proposition 1.14. $\square$

In Seydewitz’s construction, if we assume that the points R and L coincide, then we get the following:

Proposition 3.4

Let R be a point of ${{\mathrm{PG}}}(3,{\mathbf {F}})$. Let $\Phi : {{{\mathcal {S}}}}_R \longrightarrow {{{\mathcal {S}}}}^*_R$ be a projectivity. The set $\Gamma$ of points of intersection of corresponding elements under $\Phi$ is one of the following:

• the point R $( \text{ e.g., } {\mathbf {F}}={\mathbb {R}})$,

• a line through R, a plane through R, two distinct planes through R,

• a cone with vertex the point R and base a non-empty, non-degenerate conic in a plane not through R,

• a degenerate symplectic geometry ${{{\mathcal {P}}}}_r$, for some line r through R.

Proof

The set of points of intersection of corresponding elements under $\Phi$ is a cone with vertex the point R projecting the set of absolute points of a linear correlation of a plane $\pi$ not through the point R. Hence, the assertion follows from the previous proposition. $\square$

In Steiner’s construction, if the lines r and $\ell$ either intersect at a point V or coincide, then we get the following:

Proposition 3.5

Let r and $\ell$ be two lines s.t. $r\cap \ell = \{V\}$. Let $\Phi : {{{\mathcal {P}}}}_r \longrightarrow {{{\mathcal {P}}}}_{\ell }$ be a projectivity. The set of points of intersection of corresponding planes under $\Phi$ is one of the following:

• a pair of distinct planes,

• a cone with vertex the point V and base a non-empty, non-degenerate conic in a plane not through V,

Proposition 3.6

Let r be a line and let $\Phi : {{{\mathcal {P}}}}_r \longrightarrow {{{\mathcal {P}}}}_{r}$ be a projectivity. The set of points of intersection of corresponding planes under $\Phi$ is one of the following:

• the line r,

• a plane,

• a pair of distinct planes,

• the degenerate symplectic geometry ${{{\mathcal {P}}}}_r$.

Remark 3.7

If ${\mathbf {F}}$ is either an algebraically closed field or a finite field, then Seydewitz’s construction in ${{\mathrm{PG}}}(3,{\mathbf {F}})$ gives all the possible quadrics of ${{\mathrm{PG}}}(3,{\mathbf {F}})$ and also the degenerate symplectic geometry ${{{\mathcal {P}}}}_r$, for some line r. If ${\mathbf {F}}={\mathbb {R}}$ is the field of real numbers, then the only missing quadric, up to projectivities, is the quadric with equation $x_1^2+x_2^2+x_3^2+x_4^2=0$, that gives as set of points in ${{\mathrm{PG}}}(3,{\mathbf {F}})$, the empty set. With Steiner’s construction in ${{\mathrm{PG}}}(3,{\mathbf {F}})$, we miss also the elliptic quadric.

Note that also the converse holds:

Proposition 3.8

Let $\Gamma$ be either a, non-empty, quadric or a degenerate symplectic geometry of ${{\mathrm{PG}}}(3,{\mathbf {F}})$, ${\mathbf {F}}$ being a field. There exists two point R and L of $\Gamma$ and a projectivity $\Phi :{{{\mathcal {S}}}}_R \longrightarrow {{{\mathcal {S}}}}^*_L$ s.t. $\Gamma$ is the set of points of intersection of corresponding elements under $\Phi$.

4 ${\sigma }$-quadrics of rank 3 in ${{\mathrm{PG}}}({3,q^n)}$

Let $\Gamma :X^t A X^\sigma =0$ be a $\sigma$-quadric of ${{\mathrm{PG}}}(3,q^n)$ associated to a $\sigma$-sesquilinear form $\langle ~,~ \rangle .$ In this section, we assume throughout that $rk(A)=3$. Therefore, the radicals $V^\perp$ and $V^{\top }$ are one-dimensional vector subspace spaces of V, so they are points of ${{\mathrm{PG}}}(3,q^n)$. We distinguish several cases:

1) $V^\perp \ne V^{\top }$.

We may assume w.l.o.g. that the point $R=(1,0,0,0)$ is the right radical and the point $L=(0,0,0,1)$ is the left radical. It follows that

$$\begin{aligned} A=\left( \begin{array}{cccc} 0 &{} a_{12} &{} a_{13} &{} a_{14} \\ 0 &{} a_{22} &{} a_{23} &{} a_{24} \\ 0 &{} a_{32} &{} a_{33} &{} a_{34} \\ 0 &{} 0 &{} 0 &{} 0 \end{array} \right) , \end{aligned}$$

and

$$\begin{aligned}&\Gamma :(a_{12} x_1+ a_{22} x_2+ a_{32}x_3)x_2^\sigma +(a_{13} x_1+ a_{23} x_2+ a_{33}x_3)x_3^\sigma \\&\quad + (a_{14} x_1+ a_{24} x_2+ a_{34}x_3)x_4^\sigma =0. \end{aligned}$$

The degenerate collineation

$$\begin{aligned} \top : Y \in {{\mathrm{PG}}}(3,q^n) {\setminus } \{R\} \mapsto X^t A Y^\sigma =0 \in {{\mathrm{PG}}}(3,q^n)^* \end{aligned}$$

associated to the sesquilinear form maps points into planes through the point L. Points that are on a common line through R are mapped into the same plane through L. Therefore, $\top$ induces a collineation $\Phi : {{{\mathcal {S}}}}_R \longrightarrow {{{\mathcal {S}}}}^*_L$. Let

$$\begin{aligned}&{{{\mathcal {S}}}}_R=\{\ell _{\alpha ,\beta ,\gamma }:(\alpha ,\beta ,\gamma ) \in {{\mathrm{PG}}}(2,q^n)\}, \text{ where }\\&\quad \ell _{\alpha ,\beta ,\gamma }: \left\{ \begin{aligned} x_1&= \lambda \\ x_2&= \mu \alpha \\ x_3&= \mu \beta \\ x_4&= \mu \gamma \end{aligned}, (\lambda ,\mu ) \in {{\mathrm{PG}}}(1,q^n)\right. \end{aligned}$$

and

$$\begin{aligned} {{{\mathcal {S}}}}^*_L=\{\pi _{\alpha ',\beta ',\gamma '}: (\alpha ',\beta ',\gamma ') \in {{\mathrm{PG}}}(2,q^n)\}, \text{ where } \pi _{\alpha ',\beta ',\gamma '}: \alpha ' x_1+\beta 'x_2+\gamma ' x_3=0.\ \end{aligned}$$

The collineation $\Phi$ is given by $\Phi (\ell _{\alpha ,\beta ,\gamma }) = \pi _{\alpha ',\beta ',\gamma '}$, with

$$\begin{aligned} (\alpha ',\beta ',\gamma ')_t = A' (\alpha , \beta ,\gamma )_t^\sigma , \end{aligned}$$

where $A'$ is the matrix obtained by A by deleting the last row and the first column. Note that $|A'|\ne 0$ since $rk(A)=3$. It is easy to see that $\Gamma$ is the set of points of intersection of corresponding elements under the collineation $\Phi$.

If $Y=(y_1,y_2,y_3,y_4)$ is a point of $\Gamma {\setminus } \{R\}$, then the tangent plane to $\Gamma$ at the point Y is the plane $\pi _Y = Y^{\top }$ with equation $X^t A Y^\sigma =0$. It follows that for every point Y of $\Gamma {\setminus } \{R\}$ the plane $\pi _Y$ contains the point L. The tangent plane $\pi _L=L^{\top }$ to $\Gamma$ at the point L is the plane with equation $X^t A L^\sigma =0$, that is:

$$\begin{aligned} \pi _L:a_{14} x_1+a_{24}x_2 +a_{34} x_3=0. \end{aligned}$$

We again distinguish some cases.

• First, assume that $\pi _L$ contains the line RL.

  It follows that, w.l.o.g., we may put $\pi _L:x_3=0$. Hence, $a_{14}=a_{24}=0$ and we can put $a_{34}=1$, obtaining

  $$\begin{aligned} \Gamma :(a_{12} x_1+ a_{22} x_2+ a_{32}x_3)x_2^\sigma +(a_{13} x_1+ a_{23} x_2+ a_{33}x_3)x_3^\sigma + x_3x_4^\sigma =0. \end{aligned}$$

  With this assumption, the collineation $\Phi$ maps the line LR into the plane $\pi _L$. Consider now the pencil ${{{\mathcal {P}}}}_{R,\pi _L}$ of lines through R in $\pi _L$. We distinguish two cases.

i) $\Phi$ maps the lines of ${{{\mathcal {P}}}}_{R,\pi _L}$ into the planes through the line RL.

In this case, we can assume that $\Phi$ maps the line $x_3=x_4=0$ into the plane $x_2=0$ and the line $x_2=x_4=0$ into the plane $x_1=0$ obtaining

$$\begin{aligned} \Gamma : ax_1x_3^\sigma +b x_2^{\sigma +1}+x_3x_4^\sigma =0. \end{aligned}$$

We can assume that $\Gamma$ contains the points $(0,1,-1,1)$ and $(1,0,1-1)$ obtaining $a=b=1$ and hence a canonical equation is given by

$$\begin{aligned} \Gamma : x_1x_3^\sigma + x_2^{\sigma +1}+x_3x_4^\sigma =0. \end{aligned}$$

Note that in this case $\Gamma$ is the set of points studied in [13], where $\Gamma$ has been called a $\sigma$-cone. In this paper, we call this set a degenerate parabolic $\sigma$-quadric with collinear vertex points R and L.

In [13] it has been proved that the following holds:

Theorem 4.1

Let $\Gamma$ be a degenerate parabolic $\sigma$-quadric ${\Gamma }$ of ${{\mathrm{PG}}}(3,q^n)$ with collinear vertex points R and L. Then $|\Gamma |=q^{2n}+q^n+1$, RL is the unique line contained in ${\Gamma }$ and $\pi _{L}$ is the unique plane that meets ${\Gamma }$ exactly in RL.

ii) $\Phi$ does not map the lines of ${{{\mathcal {P}}}}_{R,\pi _L}$ into the planes through the line RL.

In this case, there exists a plane $\pi$ containing RL such that the lines of the pencil ${{{\mathcal {P}}}}_{R,\pi }$ are mapped, under $\Phi$ into the planes through RL. Hence, there is a unique line through R (beside RL) contained in $\Gamma$. In this case, we may assume that $\Phi$ maps the line $x_3=x_4=0$ into the plane $x_1=0$ and the line $x_2=x_4=0$ into the plane $x_2=0$. Hence:

$$\begin{aligned} \Gamma : ax_1x_2^\sigma + bx_2x_3^\sigma +x_3x_4^\sigma =0. \end{aligned}$$

Assuming that $\Gamma$ contains the points $(0,1,1,-1)$ and $(1,1,-1,0)$, we get $a=b=1$ and hence a canonical equation is given by

$$\begin{aligned} \Gamma : x_1x_2^\sigma + x_2x_3^\sigma +x_3x_4^\sigma =0. \end{aligned}$$

We will call this set a hyperbolic $\sigma$-quadric with collinear vertex points R and L.

Theorem 4.2

Let $\Gamma$ be a hyperbolic $\sigma$-quadric of ${{\mathrm{PG}}}(3,q^n)$ with collinear vertex points R and L. Then $|\Gamma |=(q^{n}+1)^2$ and $\Gamma$ contains exactly three lines: the line RL, a unique other line through R and a unique other line through L.

• $\pi _L$ does not contain the line RL (or equivalently $\Phi$ does not map the line RL into a plane through the line RL).

  W.l.o.g. we may put $\pi _L:x_1=0$. In this case, there is a plane through R (not containing L), say $\pi _R$, such that the pencil of lines through R in $\pi _R$ is mapped, under $\Phi$, into the pencil of planes through RL. We may assume that $\pi _R:x_4=0$. Hence, $\Phi$ maps the lines $\ell _{\alpha ,\beta ,0}$ into the planes $\pi _{0,\beta ',\gamma '}$, so we may assume that $\Phi$ maps the line $\ell _{1,0,0}$ into the plane $\pi _{0,1,0}$ and the line $\ell _{0,1,0}$ into the plane $\pi _{0,0,1}$. Hence, the points of $\Gamma$ satisfy the equation

  $$\begin{aligned} ax_2^{\sigma +1} - b x_3^{\sigma +1} + x_1 x_4^\sigma =0. \end{aligned}$$

Assuming, w.l.o.g., that the point $(1,1,0,-1)$ belongs to $\Gamma$, we obtain $a=1$. The number of lines through R contained in $\Gamma$ depends on the number of solutions of the equation $x^{\sigma +1} = b$, and hence, it is either 0, 1, 2 or $q+1$ depending upon q even or odd and n even or odd. We distinguish several cases:

• If q is even and n is even, then there are either 0 or 1 or $q+1$ solutions giving either 0 or 1 or $q+1$ lines through R (and hence through L) contained in $\Gamma$.

• If q is even and n is odd, then there is a unique solution of the equation giving one line through R and one line through L contained in $\Gamma$.

• If q is odd and n is even, then there are either 0 or $q+1$ solutions of the equation giving either 0 or $q+1$ lines through R (and through L) contained in $\Gamma$.

• If q is odd and n is odd, then there are either 0 or 2 solutions of the equation giving either 0 or 2 lines through R (and through L) contained in $\Gamma$.

In these cases, we will call the set $\Gamma$ either an elliptic or a parabolic or a hyperbolic or a ${(q+1)}$-hyperbolic $\sigma$-quadric with vertex points R and L according to the number of lines through R contained in $\Gamma$ is either 0 or 1 or 2 or $q+1$.

If q is even and n is even, put $r =(q^n-1, q^m+1)$.

Theorem 4.3

If $\Gamma$ is an elliptic $\sigma$-quadric of ${{\mathrm{PG}}}(3,q^n)$ with vertex points R and L, then $\Gamma$ has canonical equation $x_2^{\sigma +1}-bx_3^{\sigma +1}+x_1x_4^{\sigma }=0$, with b a non-square if q is odd and $b^{(q^n-1)/r}\ne 1$ if q is even and n is even. Moreover, $|\Gamma |=q^{2n}+1$ and $\Gamma$ contains no line.

Theorem 4.4

If $\Gamma$ is a parabolic $\sigma$-quadric of ${{\mathrm{PG}}}(3,q^n)$ with vertex points R and L, then q is even and $\Gamma$ has canonical equation $x_2^{\sigma +1}-bx_3^{\sigma +1}+x_1x_4^{\sigma }=0$, where the equation $x^{\sigma +1}=b$ has a unique solution. Moreover, $|\Gamma |=q^{2n}+q^n+1$ and $\Gamma$ contains a unique line through R and a unique line through L.

Theorem 4.5

If $\Gamma$ is a hyperbolic $\sigma$-quadric of ${{\mathrm{PG}}}(3,q^n)$ with vertex points R and L, then both q and n are odd and $\Gamma$ has canonical equation $x_2^{\sigma +1}-bx_3^{\sigma +1}+x_1x_4^{\sigma }=0$, where $x^{\sigma +1}=b$ has exactly two solutions. Moreover, $|\Gamma |= q^{2n}+2q^n+1$ and $\Gamma$ contains exactly two lines through R and exactly two lines through L.

Theorem 4.6

If $\Gamma$ is a $(q+1)$-hyperbolic $\sigma$-quadric of ${{\mathrm{PG}}}(3,q^n)$ with vertex points R and L, then n is even and $\Gamma$ has canonical equation $x_2^{\sigma +1}-bx_3^{\sigma +1}+x_1x_4^{\sigma }=0$, where $x^{\sigma +1}=b$ has exactly $q+1$ solutions. Moreover, $|\Gamma |= q^{2n}+ (q+1)q^n+1$ and $\Gamma$ contains exactly $q+1$ lines through R and exactly $q+1$ lines through L.

2) $V^\perp = V^{\top }$.

We may assume w.l.o.g. that the point $R=L=(1,0,0,0)$ is both the left radical and the right radical. It follows that, in this case, $\Gamma$ is a cone with vertex the point R. Since the matrix A has rank three with first column and last row equal to 0, by choosing a plane not through the point R, e.g., $\pi :x_1=0$, we get that the set $\Gamma \cap \pi$ is a $\sigma$-conic of the plane $\pi$ with associated matrix of rank 3. Hence, it is a Kestenband $\sigma$-conic of $\pi$. It follows that $\Gamma$ is a cone with vertex the point R projecting a Kestenband $\sigma$-conic in a plane not through R.

5 ${\sigma }$-quadrics of rank at most 2 in ${{\mathrm{PG}}}({3,q^n)}$

In this section, a $\sigma$-quadric $\Gamma$ of ${{\mathrm{PG}}}(3,q^n)$ will have equation $X^t A X^\sigma =0$ with $rk(A) \le 2$.

5.1 $\sigma$-quadrics of rank 2

Since $rk(A)=2$, it follows that the left and right radicals are two lines r and $\ell$ of ${{\mathrm{PG}}}(3,q^n)$. We distinguish three cases.

1) $r\cap \ell = \emptyset .$

We may assume w.l.o.g. that $r:x_3=x_4=0$ and $\ell :x_1=x_2=0$. Then:

$$\begin{aligned} \Gamma : (a_{31}x_3+a_{23}x_2)x_3^\sigma +(a_{14}x_1+a_{24}x_2)x_4^\sigma =0, \end{aligned}$$

that is the set of points of ${{\mathrm{PG}}}(3,q^n)$ of intersection of corresponding planes under a collineation $\Phi : {{{\mathcal {P}}}}_r\longrightarrow {{{\mathcal {P}}}}_{\ell }$, where

$$\begin{aligned} {{{\mathcal {P}}}}_r= & {} \{\pi _{a,b}:ax_3+bx_4=0\}_{\{(a,b)\in {{\mathrm{PG}}}(1,q^n)\}},\\ {{{\mathcal {P}}}}_{\ell }= & {} \{\pi '_{a,b}:ax_1+bx_2=0\}_{\{(a,b)\in {{\mathrm{PG}}}(1,q^n)\}}. \end{aligned}$$

The set $\Gamma$ contains the $q^n+1$ lines of a pseudoregulus (see [30]) with transversal lines r and $\ell$ We call this set a $\sigma$-quadric of pseudoregulus type with skew vertex lines r and $\ell$.

Note that if $n=2$, $\sigma ^2=1$ a pseudoregulus have been already introduced in [15] and $\sigma$-quadrics of pseudoregulus type with skew vertex lines have been introduced in [11] where they were called hyperbolic $Q_F$-sets.

Let $\Phi : {{{\mathcal {P}}}}_r \longrightarrow {{{\mathcal {P}}}}_s$ be a collineation with accompanying automorphism $\sigma : x\mapsto x^{q^m}$, $(m,n)=1$.

Assuming that :

$$\begin{aligned} \Phi (\pi _{1,0}) = \pi '_{1,0}, \Phi (\pi _{0,1}) = \pi '_{0,1}, \Phi (\pi _{1,1}) = \pi '_{1,1}, \end{aligned}$$

we have that $\Phi (\pi _{a,b}) = \pi '_{a^\sigma ,b^\sigma }$.

Hence, the set ${{{\mathcal {Q}}}}$ of points of intersection of corresponding planes under $\Phi$ is given by the points whose homogeneous coordinates are the solutions of the linear system

$$\begin{aligned} \left\{ \begin{aligned}&ax_3+bx_4 = 0 \\&a^{\sigma } x_1+b^{\sigma } x_2 = 0, \end{aligned} \right. \end{aligned}$$

where $(a,b) \in {{\mathrm{PG}}}(1,q^n)$. This system is equivalent to the linear system

$$\begin{aligned} \left\{ \begin{aligned}&ax_3+bx_4 = 0 \\&a x_1^{{\sigma }^{-1}}+b x_2^{{\sigma }^{-1}} =0. \end{aligned} \right. \end{aligned}$$

A point $P=(x_1,x_2,x_3,x_4)$ belongs to ${{{\mathcal {Q}}}}$ if and only if the previous linear system, in the unknowns a and b, has non-trivial solutions, hence if and only if $x_1^{\sigma ^{-1}}x_4-x_2^{\sigma ^{-1}}x_3=0$, that is $x_1x_4^\sigma -x_2x_3^\sigma =0.$ This can be seen as a canonical equation of a $\sigma$-quadric of pseudoregulus type with skew vertex lines. Let $c\in {{\mathbb {F}}}_q^*$ and let $\gamma \in {{\mathbb {F}}}_{q^n}^*$ be such that $N(\gamma )=c$. Put

$$\begin{aligned} {{{\mathcal {Q}}}}_c =\{(\gamma x^{\sigma }, \gamma y^{\sigma },x,y)\}_{(x,y)\in {{\mathrm{PG}}}(1,q^n)}. \end{aligned}$$

The set ${{{\mathcal {Q}}}}_c$ is a maximum scattered linear set of rank 2n of ${{\mathrm{PG}}}(3,q^n)$ of pseudoregulus type with transversal lines r and $\ell$. Hence, the set ${{{\mathcal {Q}}}}$ is the union of the skew lines r, $\ell$ and the $q-1$ linear sets of pseudoregulus type ${{{\mathcal {Q}}}}_c$, $c \in {{\mathbb {F}}}_{q^n}^*$.

The following holds:

Proposition 5.1

Let $\Gamma$ be a $\sigma$-quadric of pseudoregulus type of ${{\mathrm{PG}}}(3,q^n)$ with skew transversal lines r and $\ell$. The only lines contained in $\Gamma$ are $r, \ell$ and the $q^n+1$ lines of the pseudoregulus associated to $\Gamma$.

Proposition 5.2

Let a, b and c be three pairwise skew lines and let $\ell$ and s be two skew lines meeting a, b and c. There is a unique $\sigma$-quadric of pseudoregulus type of ${{\mathrm{PG}}}(3,q^n)$ with skew vertex lines r and $\ell$ containing a, b and c.

2) $r \cap \ell =\{V\}$ is a point.

We may assume w.l.o.g. that $r:x_3=x_4=0, \ell :x_2=x_3=0$. In this case, the $\sigma$-quadric $\Gamma$ is a cone with vertex the point V projecting a (degenerate or not) $C_F^m$-set in a plane not through V. Indeed, let $\pi$ be a plane not through the point V and let $R=r\cap \pi$, $L=\ell \cap \pi$. It follows that $\Gamma \cap \pi$ is a set of points of $\pi$ generated by a collineation between the pencils of lines of $\pi$ with center the points R and L induced by the collineation between the pencil of planes ${{{\mathcal {P}}}}_r$ and ${{{\mathcal {P}}}}_{\ell }$ that is associated to $\Gamma$.

3) $r=\ell$.

We may assume w.l.o.g. that $r=\ell :x_3=x_4=0$. In this case, the $\sigma$-quadric is a cone with vertex the line r over a $\sigma$-quadric of a line skew with r. That is $\Gamma$ is either just the line r or a plane through r or a pair of distinct planes through r or $q+1$ planes through r forming an ${{\mathbb {F}}}_q$-subpencil of planes through r.

5.2 ${\sigma }$-quadrics of rank 1

In this subsection, a $\sigma$-quadric $\Gamma$ will have equation $X^t A X^\sigma =0$ with $rk(A)=1$. Hence, dim$V^\perp =$ dim $V^{\top }=3$ so left and right radicals in ${{\mathrm{PG}}}(3,q^n)$ are planes. We distinguish two cases:

• $V^\perp \ne V^{\top }$.

  We may assume that $r:x_4=0$ is the right radical and $\ell :x_1=0$ is the left radical. Hence, $\Gamma :x_1x_4^\sigma =0$, that is the union of two different planes.

• $V^\perp =V^{\top }$.

  We may assume $r=\ell :x_4=0$ is both the left and right radical. Hence, $\Gamma :x_4^{\sigma +1}=0$, that it is a plane of ${{\mathrm{PG}}}(3,q^n)$.

6 $\sigma$-quadrics of ${{\mathrm{PG}}}(3,q^n)$ with $|A|=0$

In this section, we summarize the results obtained on $\sigma$-quadrics.

Proposition 6.1

Let $\Gamma :X^t A X^\sigma =0$ be a $\sigma$-quadric of ${{\mathrm{PG}}}(3,q^n$), with $|A|=0$. The set $\Gamma$ is one of the following:

• a cone with vertex a line v projecting a $\sigma$-quadric of a line $\ell$ skew with v (hence either just the line v or one, two or $q+1$ planes through v),

• a cone with vertex a point V projecting either a Kestenband $\sigma$-conic or a (possibly degenerate) $C_F^m$-set of a plane $\pi$, with $V\notin \pi$,

• a degenerate either parabolic or hyperbolic $\sigma$-quadric with two collinear vertex points,

• a non-degenerate either elliptic or parabolic or hyperbolic or $(q+1)$-hyperbolic $\sigma$-quadric with two vertex points,

• a non-degenerate $\sigma$-quadric with two skew vertex lines (i.e., a $\sigma$-quadric of pseudoregulus type).

Let $\sigma :x\mapsto x^{q^m}$, $\sigma ':x\mapsto x^{q^{m'}}$, $(m,n)=(m',n)=1$.

Proposition 6.2

Let $\Gamma$ be a $\sigma$-quadric and let $\Gamma '$ be a $\sigma '$-quadric of ${{\mathrm{PG}}}(3,q^n)$. If $\Gamma$ and $\Gamma '$ are cones, then $\Gamma$ and $\Gamma '$ are ${\mathrm {P}}\Gamma {}{\mathrm {L}}$-equivalent if and only if the basis of the cones are ${\mathrm {P}}\Gamma {}{\mathrm {L}}$-equivalent.

We will say that a $\sigma$-quadric and a $\sigma '$-quadric of ${{\mathrm{PG}}}(3,q^n)$ are of the same type if they have the same name.

Proposition 6.3

Let $\Gamma$ be a $\sigma$-quadric and let $\Gamma '$ be a $\sigma '$-quadric of ${{\mathrm{PG}}}(3,q^n)$. If $\Gamma$ and $\Gamma '$ are not cones and have non-empty vertices, then $\Gamma$ and $\Gamma '$ are ${\mathrm {P}}\Gamma {}{\mathrm {L}}$-equivalent if and only if either $\sigma '=\sigma$ or $\sigma ' = \sigma ^{-1}$ and $\Gamma$, $\Gamma '$ are of the same type.

Proof

Since $\Gamma$ and $\Gamma '$ are not cones, both $\Gamma$ and $\Gamma '$ have two distinct vertices. We divide two cases.

Case 1. The vertices of both $\Gamma$ and $\Gamma '$ are points.

We may assume that $\Gamma$ and $\Gamma '$ have the same vertices R and L. Let $\Phi$ (resp. $\Phi '$) be the $\sigma$-collineation (resp. $\sigma '$-collineation) between the star of lines with center R and the star of planes with center L associated to ${\Gamma }$ (resp. ${\Gamma }'$). Observe that ${\Gamma }$ is also a $\sigma ^{-1}$-quadric with vertices L and R to which there is associated the $\sigma ^{-1}$-collineation $\Phi ^{-1}$.

Let f be a collineation of ${{\mathrm{PG}}}(3,q^n)$ mapping ${\Gamma }$ into ${\Gamma }'$. Since the number of lines of $\Gamma$ through either R or L is at most $q+1$, there exists plane $\pi$ through the line RL s.t. the lines of $\pi$ through R are mapped under $\Phi$ onto planes through a line on L not contained in $\pi$. It follows that $\Phi$ induces a collineation $\pi$ between the pencil of lines through R and the pencil of lines through L. This gives that $\Gamma \cap \pi$ is a (possibly degenerate) $C_F^m$-set of $\pi$. It follows that $f(\Gamma \cap \pi )$ is a (possibly degenerate) either $C_F^m$ or $C_F^{n-m}$-set. Hence, $\sigma '=\sigma$ or $\sigma '=\sigma ^{-1}$.

Case 2. The vertices of both $\Gamma$and $\Gamma '$ are two skew lines.

We may assume that $\Gamma$ and $\Gamma '$ have the same vertices r and $\ell$. Every plane not through r neither through $\ell$ meets $\Gamma$ in a (possibly degenerate) $C_F^m$ set. The assertion follows in the same way as in the previous case. $\square$

7 ${\sigma }$-quadrics of rank 4 of ${{\mathrm{PG}}}({d,q^n})$

Opposite to the case of ${{\mathrm{PG}}}(2,q^n)$, nothing is known for the sets of absolute points of ${{\mathrm{PG}}}(d,q^n)$ of a nonlinear correlation, different from a polarity, of ${{\mathrm{PG}}}(d,q^n)$, $d\ge 3$. We recall that took roughly 14 years and 10 different papers to B. Kestenband to classify all $\sigma$-conics of ${{\mathrm{PG}}}(2, q^n)$ with equation $X^t A X^\sigma =0$, $|A|\ne 0$. In this section, $\Gamma$ will be a $\sigma$-quadric of ${{\mathrm{PG}}}(d,q^n)$ with equation $X^t A X^\sigma =0$, $|A|\ne 0$. We determine the dimension of the maximum totally isotropic subspaces contained in $\Gamma$.

Proposition 7.1

If $S_h$ is an h-dimensional subspace of ${{\mathrm{PG}}}(d,q^n)$ contained in a $\sigma$-quadric $\Gamma$ with equation $X^t A X^\sigma =0$, $|A|\ne 0$, then $h\le \left\lfloor {\frac{d-1}{2}}\right\rfloor$. Moreover, there exists a $\sigma$-quadric with equation $X^t A X^\sigma =0$, $|A| \ne 0$, containing a subspace with dimension $\left\lfloor {\frac{d-1}{2}}\right\rfloor .$

Proof

Let S be a subspace with maximum dimension, say h, contained in $\Gamma$. We may assume, w.l.o.g., that S has equations $x_{h+2} = x_{h+3} =\ldots x_{d+1}=0$. It follows that the matrix A has the submatrix formed by the first $h+1$ rows and $h+1$ columns with entries all 0’s. This gives that $d+1={\mathrm {rank}}(A)\le 2(d-h)$, and hence $h\le \left\lfloor {\frac{d-1}{2}}\right\rfloor$. If d is odd, then the $\sigma$-quadric with equation $x_1{x_{d+1}}^{\sigma }+x_2{x_d}^{\sigma }+\ldots +x_{d+1}{x_1}^{\sigma }=0$ contains the $\frac{d-1}{2}$-dimensional subspace with equations $x_1=x_2=\ldots =x_{\frac{d+1}{2}} =0$. If d is even, then the $\sigma$-quadric with equation $x_1{x_{d+1}}^{\sigma }+x_2{x_d}^{\sigma }+ \ldots +x_{\frac{d}{2}+1}^{\sigma +1}+ \ldots +x_{d+1}{x_1}^{\sigma }=0$ contains the $(\frac{d}{2}-1)$-dimensional subspace with equations $x_1=x_2=\ldots =x_{\frac{d}{2}+1} =0$. $\square$

A similar proof can be given to obtain the following, more general, result

Proposition 7.2

If $S_h$ is an h-dimensional subspace of ${{\mathrm{PG}}}(d,q^n)$ contained in a $\sigma$-quadric $\Gamma$ with equation $X^t A X^\sigma =0$, then $h\le \left\lfloor {d-\frac{{\mathrm {rank}}(A)}{2}}\right\rfloor$. Moreover, there exists a $\sigma$-quadric with equation $X^t A X^\sigma =0$, containing a subspace with dimension $\left\lfloor {d-\frac{{\mathrm {rank}}(A)}{2}}\right\rfloor$.

8 Applications of $\sigma$-quadrics of ${{\mathrm{PG}}}(3,2^n)$

8.1 $\sigma$-quadrics of pseudoregulus type and the Segre’s $(2^n+1)$-arc of ${{\mathrm{PG}}}(3,2^n)$,

An arc in ${{\mathrm{PG}}}(3, q)$ is a set of points with the property that any 4 of them span the whole space. The maximum size of an arc in ${{\mathrm{PG}}}(3,q)$ is $q+1$ as proved by Segre in [34] for q odd and by Casse in [6] for q even. An example of $(q+1)$-arc of ${{\mathrm{PG}}}(3,q)$, $q=2^n$, was given by Segre in [35] and it is a set of points projectively equivalent to the set $\{ (1,t,t^\sigma ,t^{\sigma +1})\} \cup \{(0,0,0,1)\}$, where $\sigma : x\mapsto x^{2^m}$, $(m,n)=1$. The $(q+1)$-arcs of ${{\mathrm{PG}}}(3,q)$ have been classified by Segre for q odd, $q\ge 5$ [34] and by Casse and Glynn for q even, $q\ge 8$ [7] and are the twisted cubic and, for q even, the Segre’s $(q+1)$-arc.

An ovoid of $Q^+(3,q)$ is a set of $q+1$ points no two collinear on the quadric. A translation ovoid is an ovoid ${{{\mathcal {O}}}}$ containing a point P s.t. there is a collineation group of $Q^+(3,q)$ fixing P linewise and acting sharply transitively on the points of ${{{\mathcal {O}}}}{\setminus } \{P\}$. An example of translation ovoid of $Q^+(3,q)$ is a non degenerate conic.

Proposition 8.1

Let $\Gamma$ be a $\sigma$-quadric of pseudoregulus type of ${{\mathrm{PG}}}(3,q^n)$ with skew vertex lines r and $\ell$. The set $\Gamma$ is mapped, via the Klein correspondence, into the union of a translation ovoid of a hyperbolic quadric ${\mathcal {Q}}^+(3,q^n)$ with two points R and S. Moreover, if $q=2$, then this translation ovoid is the Segre’s $(2^n+1)$-arc of ${{\mathrm{PG}}}(3,2^n)$.

Proof

Let $r:x_3=x_4=0$ and $\ell :x_1=x_2=0$ and let ${\Gamma }$ be the $\sigma$-quadric with vertex lines $\ell$ and r with equation $x_1x_4^\sigma - x_2x_3^\sigma =0$. The lines of the associated pseudoregulus are spanned by the points $(0,0,\alpha ,\beta )$ and $(c\alpha ^\sigma ,c\beta ^\sigma ,0,0)$, $(\alpha ,\beta )\in {{\mathrm{PG}}}(1,q^n)$, $c\in {{\mathbb {F}}}_q^*$. Via the Klein correspondence, the lines of the pseudoregulus are mapped to the set of points $\{ (0,\alpha ^{\sigma +1},\alpha ^\sigma \beta , \alpha \beta ^\sigma ,\beta ^{\sigma +1},0)\}_{\{(\alpha ,\beta )\in {{\mathrm{PG}}}(1,q^n)\}}$, i.e., the set of points of the ovoid

$$\begin{aligned} {{{\mathcal {O}}}}=\{(0,\alpha ^{\sigma +1},\alpha ^\sigma ,\alpha ,1,0)\} \cup \{P_\infty =(0,0,0,0,1,0)\} \end{aligned}$$

of the hyperbolic quadric ${\mathcal {Q}}^+(3,q^n)$ with equations $x_1=x_6=0, x_2x_5-x_3x_4=0$ contained in the Klein quadric with equation $x_1x_6-x_2x_5+x_3x_4=0$. The two vertex lines r and $\ell$ are mapped, via the Klein correspondence, into the two points (0, 0, 0, 0, 0, 1) and (1, 0, 0, 0, 0, 0) on the line $x_2=x_3=x_4=x_5=0$ that is the polar line w.r.t. the three-dimensional subspace with equations $x_1=x_6=0$ containing the ovoid ${{{\mathcal {O}}}}$ under the polarity defined by the Klein quadric.

The ovoid ${{{\mathcal {O}}}}$ is a translation ovoid w.r.t. the point $P_\infty$ of the hyperbolic quadric ${\mathcal {Q}}^+(3,q^n)$ since the group of projectivities of ${{\mathrm{PG}}}(5,q^n)$ induced by the matrices

$$\begin{aligned} \left( \begin{array}{cccccc} 1 &{} 0 &{} 0 &{} 0&{}0 &{}0 \\ 0 &{} 1 &{} 0 &{} 0&{}0&{}0\\ 0 &{} b &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} b^\sigma &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} b^{\sigma +1} &{} b^\sigma &{} b &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{}0 &{}0 &{} 1 \end{array} \right) \end{aligned}$$

$b\in {{\mathbb {F}}}_{q^n}$, stabilizes $P_\infty$ and acts sharply transitively on the points of ${{{\mathcal {O}}}}{\setminus } \{P_\infty \}$. $\square$

8.2 Degenerate $\sigma$-quadrics and Lüneburg spread of ${{\mathrm{PG}}}(3,2^n)$.

In 1965, H. Lüneburg proved that if $q^n=2^{2h+1}$, $h\ge 1$, then the set of absolute lines of a polarity of $W(3,q^n)$ is a symplectic spread, now called the Lüneburg spread of ${{\mathrm{PG}}}(3,q^n)$ (see [32]).

Let $\Sigma _{\infty }$ be a hyperplane of ${{\mathrm{PG}}}(4,q^n)$ and let ${\mathcal {Q}}^+(3,q^n)$ be a hyperbolic quadric of $\Sigma _{\infty }$. A set ${{{\mathcal {A}}}}$ of $q^{2n}$ points of ${{\mathrm{PG}}}(4,q^n) {\setminus } \Sigma _{\infty }$ s.t. the line joining any two of them is disjoint from ${\mathcal {Q}}^+(3,q^n)$ is called an affine set of ${{\mathrm{PG}}}(4,q^n) {\setminus } \Sigma _{\infty }$. In what follows, we will denote by $\perp$ the polarity of ${\mathcal {Q}}^+(5,q^n)$. In [29] and also in [31] the following result has been proved.

Theorem 8.2

Let ${{{\mathcal {O}}}}$ be an ovoid of ${\mathcal {Q}}^+(5,q^n)$, let P be a point of ${{{\mathcal {O}}}}$ and let $\Omega$ be a hyperplane of ${{\mathrm{PG}}}(5,q^n)$ not containing P. The set ${{{\mathcal {A}}}}_P({{{\mathcal {O}}}})$ obtained by projecting ${{{\mathcal {O}}}}$ from the point P into $\Omega$ is an affine set of $\Omega {\setminus } P^{\perp }$. Conversely, if ${{{\mathcal {A}}}}$ is an affine set of $\Omega {\setminus } P^{\perp }$, then the set ${{{\mathcal {O}}}}= \{PR \cap {\mathcal {Q}}^+(5,q^n) : R \in {{{\mathcal {A}}}} \}$ is an ovoid of ${\mathcal {Q}}^+(5,q^n)$.

If ${{{\mathcal {S}}}}$ is a spread of ${{\mathrm{PG}}}(3,q^n)$ and $\ell$ is a line of ${{{\mathcal {S}}}}$, then we will denote by ${{{\mathcal {A}}}}_{\ell }({{{\mathcal {S}}}})$ the affine set arising from ${{{\mathcal {S}}}}$ with respect to $\ell$.

If ${{{\mathcal {S}}}}$ is a symplectic spread, then ${{{\mathcal {A}}}}_{\ell }({{{\mathcal {S}}}})$ is a set of $q^{2n}$ points of an affine space ${{\mathrm{PG}}}(3,q^n) {\setminus } \pi _{\infty }$ such that the line joining any two of them is disjoint from a given non-degenerate conic ${{{\mathcal {C}}}}$ of $\pi _{\infty }$.

The affine set arising from the Lüneburg spread has been studied by A. Cossidente, G. Marino and O. Polverino in [8], where the following result has been obtained:

Theorem 8.3

The affine set ${{{\mathcal {A}}}}$ of the Lüneburg spread of ${{\mathrm{PG}}}(3,2^{2h+1})$ is the union of $2^{2h+1}$-arcs, and each of them can be completed to a translation hyperoval. The directions of ${{{\mathcal {A}}}}$ on $\pi _\infty$ are the complement of a regular hyperoval.

With the following theorem, we observe that the affine set of a Lüneburg spread of ${{\mathrm{PG}}}(3,2^{2h+1})$ is the affine part of a degenerate elliptic $\sigma$-quadric of ${{\mathrm{PG}}}(3,2^{2h+1})$ with two vertex points.

Theorem 8.4

Let ${\Gamma }$ be a degenerate parabolic $\sigma$-quadric of ${{\mathrm{PG}}}(3,2^n)$ with collinear vertex points R and L. The lines joining any two points of ${\Gamma } {\setminus } RL$ are disjoint from a translation hyperoval ${{{\mathcal {O}}}}_{\infty }$ of the plane ${\pi }_{L}$, projectively equivalent to the set $\{ (0,t,t^{{\sigma }^{-2}},1) : t \in {{\mathbb {F}}}_{2^n} \} \cup \{ R, L \}$.

If $n=2h+1$ and $\sigma$ is the automorphism of ${{\mathbb {F}}}_{2^n}$ given by $\sigma : x \mapsto x^{2^h}$, then ${{{\mathcal {O}}}}_{\infty }$ is a regular hyperoval. Hence, the set ${\Gamma }{\setminus } RL$ is the affine set of the Lüneburg spread of ${{\mathrm{PG}}}(3,q^n).$

Proof

Let ${\Gamma }$ be a degenerate parabolic $\sigma$-quadric with collinear vertex points $R=(0,0,1,0)$ and $L=(0,1,0,0)$ with equation

$$\begin{aligned} {x_4}^{\sigma +1}+x_1x_2^{\sigma }-x_3{x_1}^{\sigma }=0. \end{aligned}$$

It follows that the plane $\pi _{RL}$ has equation $x_1=0$. The set ${{{\mathcal {A}}}} = {{{\mathcal {K}}}}{\setminus } \pi _{RL}$ is given by ${{{\mathcal {A}}}}=\{(1,x,x^\sigma +y^{\sigma +1},y):x,y \in {{\mathbb {F}}}_{q^n}\}$. Arguing as in Proposition 5.2 in [8], we obtain that the set of directions determined by ${{{\mathcal {A}}}}$ into the plane $\pi _{RL}$ covers all the points of $\pi _{RL}$ except to the points of a hyperoval ${{{\mathcal {C}}}}$ given by ${{{\mathcal {C}}}}=\{(0,x,x^{\sigma ^{-2}},1):x \in {{\mathbb {F}}}_{q^n}^*\}$. Note that if $q=2^{2h+1}$ and $\sigma :x\mapsto x^{2^h}$, then the hyperoval ${{{\mathcal {C}}}}$ is a hyperconic. The assertion follows (see Sect. 8.2). $\square$