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

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 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 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 PG ( 3 , 2 n ) and to the Lüneburg spread of PG ( 3 , 2 2 h + 1 ) .


Remark 1.5
A (possibly degenerate) correlation induced by a σ -sesquilinear form will be also called a σ -correlation. Sometimes we will call linear a (degenerate) correlation whose associated form is a bilinear form.

Reflexive sesquilinear forms and polarities
Definition 5 A (degenerate) polarity is a (degenerate) correlation whose square is the identity.
If ⊥ is a (possibly degenerate) polarity, then for every pair of points P and R the following holds:

Proposition 1.6 A (degenerate) correlation is a (degenerate) polarity if and only if the induced sesquilinear form is reflexive.
Proof Let , be a reflexive σ -sesquilinear form of F d+1 q . If u ∈ v ⊥ , then v ∈ u ⊥ . Hence, the map u → u ⊥ defines a (possibly degenerate) polarity. Conversely, given a (possibly degenerate) polarity ⊥, if v ∈ u ⊥ , then u ∈ v ⊥ . So, the induced sesquilinear form is reflexive.
The non-degenerate, reflexive σ -sesquilinear forms of a (d + 1)-dimensional 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 , be a non-degenerate, reflexive σ -sesquilinear form of a (d + 1)dimensional F-vector space. Then, up to a scalar factor, the form , is one of the following: (i) a symmetric form, i.e., (ii) an alternating form, i.e., (iii) a Hermitian form, i.e., Let V be a (d + 1)-dimensional F-vector space and consider the associated projective geometry PG(d, F). If V is equipped with a sesquilinear from , , we may consider in PG(d, F) the set of absolute points of the associated correlations, that is the points X such that X ∈ X ⊥ (or equivalently X ∈ X ). If A is the associated matrix to the σ -sesquilinear form , w.r.t. an ordered basis of V , then the set has equation X t AX σ = 0.
From the previous theorem, the polarities of PG(d, F) are in one to one correspondence with non-degenerate, reflexive σ -sesquilinear forms of F d+1 . Hence, to every polarity of PG(d, F) there is an associated pair (A, σ ), with A a non-singular matrix of order d + 1 and σ an automorphism of F. In what follows, we will identify a polarity with a pair (A, σ ). From the last theorem of the previous section, we have the following: Theorem 1.8 Let (A, σ ) be a polarity of PG(d, q), one of the following holds: (i) σ = 1, A is a symmetric matrix. The polarity is called an orthogonal polarity. If q is even, there is a non-absolute point. (ii) σ = 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) σ 2 = 1, so σ : x → x √ 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 σ t , σ 2 = 1, σ = 1. Each of the above polarities of PG(d, q) determines a set : X t AX σ = 0, as set of its absolute points. The three types of polarities give rise to the following, well known, subsets of PG(d, q): is a polarity of PG(d, q), then one of the following holds: (i) is called a quadric of PG(d, q) (q odd, orthogonal polarity). is a hyperplane of PG(d, q) (q even, orthogonal polarity). (ii) is the full pointset of PG(d, q) (d odd, symplectic polarity) and the geometry determined is a symplectic polar space.

Remark 1.9
If q is even, σ = 1 and (A, σ ) is an orthogonal polarity, so A is a symmetric matrix, then the set 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 is often called also non-degenerate since the associated σ -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, σ ) is a polarity of PG(d, q), then let : X t AX σ = 0 be the set of its absolute points. The following hold: (i) If is a quadric, so q is odd, then we have: (1) If d is even, then (2) If d is odd, then either If is a hyperplane, so q is even, then : If is a symplectic polar space, then is the full pointset of PG(d, q), d odd.
The geometry determined will be denoted by W (d, q). A canonical form for the associated bilinear form is (iii) If is a Hermitian variety, then

Remark 1.11
Regarding degenerate, reflexive σ -sesquilinear forms of V = F d+1 q , it is possible to prove that if dim V ⊥ = r + 1, r ≥ 0, then the set of absolute points in PG(d, q) of the associated degenerate polarity is, in all the possible cases, a cone (V r , Q d−1−r ) with vertex a subspace 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 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 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 PG(d −1−r ), then d and r must have the same parity and the cone ( 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.

Sets of the absolute points of a linear correlation of PG(d, F)
Let be the set of the absolute points of a linear correlation of PG(d, F) with equation X t AX = 0, |A| = 0. Assuming that is a proper subset of PG(d, F), the matrix A cannot be a skew-symmetric matrix and is a (possibly degenerate) quadric and vice versa. Note that the previous holds, independently of the characteristic of the field F. If the characteristic of 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 F either odd or 0, this relation is lost and, a bit surprisingly, the following holds: Moreover, since in the finite Desarguesian projective spaces all the quadrics are nonempty (see, e.g., [38]), we have the following: Proposition 1.14 If : X t AX = 0, with |A| = 0, is a set of points of PG(d, q n ), then is either a degenerate symplectic geometry or a (possibly degenerate) quadric. Vice versa if is either a degenerate symplectic geometry or a (possibly degenerate) quadric of PG(d, q n ), then there is a matrix A with |A| = 0 s.t. is projectively equivalent to the set of points with equation X t AX = 0.

F q
F q F q -linear sets of PG(d, q n ) Definition 7 Let = PG(r − 1, q n ), q = p h , p a prime. A set L is said an F q -linear set of of rank t if it is defined by the nonzero vectors of an F q -vector subspace U of V = F r q n of dimension t, that is In [2] it is proved that if L U is a scattered F q linear set of , then t ≤ rn/2. It is clear, by definition that L U is a scattered F q -linear set of rank t if and only if |L U | = q t−1 + q t−2 + · · · + q + 1. If L is a scattered linear set of PG(r − 1, q n ) of rank rn/2, it is called a maximum scattered linear set.
If dim F q U = dim F q n V = r and U F q n = V , then L U ∼ = PG(U , F q ) is a subgeometry of . In such a case, each point has weight 1, and hence |L U | = q r −1 + q r −2 + · · · + q + 1. Let = PG(t, q) be a subgeometry of = PG(t, q n ) and suppose that there exists a (t −r )-dimensional subspace of disjoint from . Let = PG(r −1, q n ) be an (r − 1)-dimensional subspace of disjoint from , and let be the projection of from to , i.e., = { , x ∩ : x ∈ }. Let p , be the map from \ to defined by x → , x ∩ . We call the center and the axis of p , . In [31] the following characterization of F q -linear sets is given: 15 If L is a projection of PG(t, q) into = PG(r − 1, q n ), then L is an F q -linear set of of rank t + 1 and L = . Conversely, if L is an F q -linear set of of rank t + 1 and L = , then either L is a subgeometry of or there are a (t − r )dimensional subspace of = PG(t, q n ) disjoint from and a subgeometry of disjoint from such that L = p , ( ).
A family of maximum scattered linear sets to which a geometric structure, called pseudoregulus, can be associated has been defined in [30].
We call the set of lines P L = {s i : i = 1, . . . , m} the F q -pseudoregulus (or simply the pseudoregulus) of associated with L and we refer to T 1 and T 2 as transversal spaces of 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 F q -linear sets of the projective line = PG(1, q t ) with a structure resembling that of an F q -linear set of PG(2n − 1, q t ), n, t ≥ 2, of pseudoregulus type has been studied. Let P 1 = w q t and P 2 = v q t be two distinct points of and let τ be an automorphism of F q t such that Fi x(τ ) = F q . For each ρ ∈ F * q t , the set W ρ,τ = {λw + ρλ τ v : λ ∈ F q t }, is an F q -vector subspace of V = F 2 q t of dimension t and L ρ,τ = L W ρ,τ is a scattered F q -linear set of .

Definition 9
In [30] the linear sets L ρ,τ have been called of pseudoregulus type and the points P 1 and P 2 the transversal points of L ρ,τ . If L ρ,τ ∩L ρ ,τ = ∅, then L ρ,τ = L ρ ,τ . [12,30]) Note that L ρ,τ = L ρ ,τ if and only if N (ρ) = N (ρ ) (where N denotes the Norm of F q t over F q ); so P 1 , P 2 and the automorphism τ 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 PG(1, q t ). All the F q -linear sets of pseudoregulus type in = PG(1, q t ), t ≥ 2, are equivalent to the linear set L 1,σ 1 under the action of the collineation group P L(2, q t ). Remark 1. 17 We observe that the pseudoregulus of 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.

The -quadrics of PG(d, q n )
In this section, we will introduce σ -quadrics of PG(d, q n ) and we will determine their intersection with subspaces.

Definition 10
A σ -quadric of PG(d, q n ) is the set of the absolute points of a (possibly degenerate) σ -correlation, σ = 1, of PG(d, q n ). A σ -quadric of PG(2, q n ) will be called a σ -conic.

Remark 2.1 Let
: X t AX σ = 0 be the set of the absolute points of a (possibly degenerate) σ -correlation of PG(d, q 2 ) and let σ : x → x q . In this case, the set is a (possibly degenerate) Hermitian variety of PG(d, q 2 ) if A is a Hermitian matrix. We have included Hermitian varieties in the definition of a σ -quadric in order to have no exceptions everywhere. Indeed, we will see that Hermitian varieties appear as intersection of σ -quadrics of PG(d, q 2 ) with subspaces.
For the remaining part of this chapter, we can assume σ = 1. Let V = F d+1 q n , let , be a degenerate σ -sesquilinear form with associated (degenerate) correlations ⊥, and let : X t AX σ = 0 be the associated σ -quadric. We will denote by L = V ⊥ and R = V , the left and right radicals of , respectively, seen as subspaces of PG(d, q n ) that will be called the vertices of . Before giving the definition of a non-degenerate σ -quadric, we prove the following: • For every point Y ∈ \R, the hyperplane Y is union of lines through Y either contained or 1-secant or 2-secant to .
In the previous equation, it is Y t AY σ = 0, since Y ∈ and Z t AY σ = 0, since Z ∈ Y . Hence, Equation (1) becomes Y t AZ σ λ + Z t AZ σ μ = 0 and four different cases occur The second part of the statement follows in a similar way.
Both these hypersurfaces have as set of F q n -rational points the set .
Inspired by the characterization of non-degenerate quadrics, Hermitian varieties and symplectic polar spaces of PG(d, q n ) (see Proposition 1.12), we define a nondegenerate σ -quadric of PG(d, q n ) by induction on the dimension d of the projective space.

Definition 11
Let be a σ -quadric of PG(d, q n ), denote by L and R the left and right radicals of the associated sesquilinear form.
(ii) is a non-degenerate σ -conic of PG(2, q n ), if the following hold: • the tangent line L to at L intersects exactly at the point L.

The -quadrics of PG(1, q n ) and of PG(2, q n )
In this subsection, we recall what is known for the σ -quadrics of PG(1, q n ) and of PG(2, q n ) with Fi x(σ ) = F q (see [9,14]).
In what follows, we will always assume that σ = 1 since if σ = 1, the set is either a (possibly degenerate) quadric or a (possibly degenerate) symplectic geometry. Moreover, since Fi x(σ ) = F q we have that σ : x → x q m , (m, n) = 1.

Proposition 2.6
A σ -quadric of PG(1, q n ) is one of the following: • the empty set, a point, two distinct points, Obviously the following holds: Proposition 2.7 Let and be a σ -quadric and σ -quadric of PG(1, q n ), respectively. The sets and are P L-equivalent if and only if | | = | |.
The sets of the absolute points of σ -correlations of 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: • a cone with vertex a point A projecting a σ -quadric of a line not through A.
• a degenerate C m F -set (i.e., the union of a line with a scattered linear set isomorphic to PG(n, q), meeting the line in a maximum scattered F q -linear set of pseudoregulus type), • a C m F -set (i.e., the union of q −1 scattered F q -linear sets each of which isomorphic to PG(n − 1, q) with two distinct points ).
• a Kestenband σ -conic Proposition 2.9 Let be a σ -conic of PG(2, q n ) with two different vertices R and L. Associated to there is a σ -collineation between the pencils of lines with vertices R and L. Moreover, is also a σ −1 conic.
Proof To the σ -conic , there is associated a σ -collineation between the pencils of lines with vertices R and L s.t. is the set of points of intersection of corresponding lines under (resp. ) (see [9]). Moreover, the σ -conic with associated the σcollineation is also a σ −1 conic with associated the σ −1 -collineation −1 .

Proposition 2.10
Let be a σ -conic of PG(2, q n ) with two different vertices R, L and let be a σ -conic of PG(2, q n ) with two different vertices R , L . The sets and are P L-equivalent if and only if either σ = σ or σ = σ −1 .
We may assume that R = R and L = L since the group P L(3, q n ) is two-transitive on points. Let (resp. ) be the σ -collineation (resp. σ -collineation) associated with (resp. ). First, assume that is C m F -set and that is a C m F -set. Let f be a collineation of PG(2, q n ) mapping into . Since R and L are the unique points of both and 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 through the point R, we have that f ( ( )) = ( ) −1 ( f ( )). As and ( ) −1 are collineations with accompanying automorphism σ and σ −1 , it follows that σ = σ −1 . Next suppose that For every line through R, we have that f ( ( )) = ( f ( )), and hence σ = σ . Next assume that is a degenerate C m F -set and that is a degenerate C m F -set and that n ≥ 3. Let f be a collineation of PG(2, q n ) mapping into . Since the line RL is the unique line contained in both and , the collineation f stabilizes RL. The directions of the sets \RL and \RL on the line RL are both F q -linear sets of pseudoregulus type with transversal points R and L. Since the transversal points of an F q -linear set of pseudoregulus type of PG(1, q n ), n ≥ 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.
Finally, the following holds: Proof We may assume that R = R since the group P L(3, q n ) is transitive on points. Obviously, if and are P L-equivalent, then | | = | |. Vice versa if | | = | |, then both and are cones with vertex the point R projecting a σ -quadric on a line not through R. The assertion follows since two σ -quadrics of are P L-equivalent if and only if have the same size (see [9]).

Remark 2.12
We can divide the σ -conics in two families: • the degenerate σ -conics: the degenerate C m F -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 not through the point V ; • the non-degenerate σ -conics: the Kestenband σ -conics and the C m F -sets. In the remaining part of this chapter, we will determine the canonical equations and some properties for the σ -quadrics of PG(3, q n ) associated to a degenerate correlation of PG(3, q n ). Let be such a σ -quadric with equation X t AX σ = 0, A a singular matrix. We will consider three separate cases according to rk(A) ∈ {1, 2, 3}.

Seydewitz's and Steiner's projective generation of quadrics of PG(3, F)
In this section, we recall the projective generation of F. Seydewitz and J. Steiner of quadrics of PG(3, F), F a field. We start with Seydewitz's construction. In what follows, if P is a point of PG(3, F) we will denote with S P the star of lines with center P and with S * P the star of planes with center P. Theorem 3.1 (F. Seydewitz [36]) Let R and L be two distinct points of PG(3, F) and let : S R −→ S * L be a projectivity. The set of points of intersection of corresponding elements under is one of the following: • If (RL) is a plane through the line R L, then is either a quadratic cone or a hyperbolic quadric Q + (3, F), • If (RL) is a plane not through the line R L, then is a non-empty, nondegenerate quadric, i.e., either an elliptic quadric Q − (3, F) or an hyperbolic quadric Q + (3, F).

Next consider Steiner's construction.
In what follows, if s is a line of PG(3, F) we will denote by P s the pencil of planes through s. • the empty set (e.g., F = R) • a point, a line, two lines, • a non-degenerate conic, • a degenerate symplectic geometry P R , for some point R.
Proof It follows immediately from Proposition 1.14.
In Seydewitz's construction, if we assume that the points R and L coincide, then we get the following: • the point R (e.g., F = 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 P r , for some line r through R.
Proof The set of points of intersection of corresponding elements under is a cone with vertex the point R projecting the set of absolute points of a linear correlation of a plane π not through the point R. Hence, the assertion follows from the previous proposition.
In Steiner's construction, if the lines r and either intersect at a point V or coincide, then we get 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 : P r −→ P r be a projectivity. The set of points of intersection of corresponding planes under is one of the following:
• the line r , • a plane, • a pair of distinct planes, • the degenerate symplectic geometry P r .

Remark 3.7
If F is either an algebraically closed field or a finite field, then Seydewitz's construction in PG(3, F) gives all the possible quadrics of PG(3, F) and also the degenerate symplectic geometry P r , for some line r . If F = R is the field of real numbers, then the only missing quadric, up to projectivities, is the quadric with equation x 2 1 + x 2 2 + x 2 3 + x 2 4 = 0, that gives as set of points in PG(3, F), the empty set. With Steiner's construction in PG(3, F), we miss also the elliptic quadric.
Note that also the converse holds:

-quadrics of rank 3 in PG(3, q n )
Let : X t AX σ = 0 be a σ -quadric of PG(3, q n ) associated to a σ -sesquilinear form , . In this section, we assume throughout that rk(A) = 3. Therefore, the radicals V ⊥ and V are one-dimensional vector subspace spaces of V , so they are points of PG(3, q n ). We distinguish several cases: 1) V ⊥ = V . 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 The degenerate collineation : Y ∈ PG(3, q n )\{R} → X t AY σ = 0 ∈ PG(3, q n ) * 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, induces a collineation : S R −→ S * L . Let The collineation is given by ( α,β,γ ) = π α ,β ,γ , with where A is the matrix obtained by A by deleting the last row and the first column. Note that |A | = 0 since rk(A) = 3. It is easy to see that is the set of points of intersection of corresponding elements under the collineation . If Y = (y 1 , y 2 , y 3 , y 4 ) is a point of \{R}, then the tangent plane to at the point Y is the plane π Y = Y with equation X t AY σ = 0. It follows that for every point Y of \{R} the plane π Y contains the point L. The tangent plane π L = L to at the point L is the plane with equation X t AL σ = 0, that is: We again distinguish some cases.
• First, assume that π L contains the line RL. It follows that, w.l.o.g., we may put π L : x 3 = 0. Hence, a 14 = a 24 = 0 and we can put a 34 = 1, obtaining With this assumption, the collineation maps the line L R into the plane π L . Consider now the pencil P R,π L of lines through R in π L . We distinguish two cases.
i) maps the lines of P R,π L into the planes through the line RL.
In this case, we can assume that 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 We can assume that contains the points (0, 1, −1, 1) and (1, 0, 1 − 1) obtaining a = b = 1 and hence a canonical equation is given by Note that in this case is the set of points studied in [13], where has been called a σ -cone. In this paper, we call this set a degenerate parabolic σ -quadric with collinear vertex points R and L.
In [13] it has been proved that the following holds: Theorem 4.1 Let be a degenerate parabolic σ -quadric of PG(3, q n ) with collinear vertex points R and L. Then | | = q 2n + q n + 1, R L is the unique line contained in and π L is the unique plane that meets exactly in R L.
ii) does not map the lines of P R,π L into the planes through the line RL. In this case, there exists a plane π containing RL such that the lines of the pencil P R,π are mapped, under into the planes through RL. Hence, there is a unique line through R (beside RL) contained in . In this case, we may assume that 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: Assuming that 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 We will call this set a hyperbolic σ -quadric with collinear vertex points R and L. • π L does not contain the line RL (or equivalently does not map the line RL into a plane through the line RL). W.l.o.g. we may put π L : x 1 = 0. In this case, there is a plane through R (not containing L), say π R , such that the pencil of lines through R in π R is mapped, under , into the pencil of planes through RL. We may assume that π R : x 4 = 0. Hence, maps the lines α,β,0 into the planes π 0,β ,γ , so we may assume that maps the line 1,0,0 into the plane π 0,1,0 and the line 0,1,0 into the plane π 0,0,1 . Hence, the points of satisfy the equation Assuming, w.l.o.g., that the point (1, 1, 0, −1) belongs to , we obtain a = 1. The number of lines through R contained in depends on the number of solutions of the equation x σ +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 . In these cases, we will call the set either an elliptic or a parabolic or a hyperbolic or a (q + 1)-hyperbolic σ -quadric with vertex points R and L according to the number of lines through R contained in 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 is an elliptic σ -quadric of PG(3, q n ) with vertex points R and L, then has canonical equation x
with b a non-square if q is odd and b (q n −1)/r = 1 if q is even and n is even. Moreover, | | = q 2n + 1 and contains no line.

Theorem 4.4 If is a parabolic σ -quadric of PG(3, q n ) with vertex points R and L, then q is even and has canonical equation x
where the equation x σ +1 = b has a unique solution. Moreover, | | = q 2n + q n + 1 and contains a unique line through R and a unique line through L.

Theorem 4.5
If is a hyperbolic σ -quadric of PG(3, q n ) with vertex points R and L, then both q and n are odd and has canonical equation , where x σ +1 = b has exactly two solutions. Moreover, | | = q 2n + 2q n + 1 and contains exactly two lines through R and exactly two lines through L.

Theorem 4.6
If is a (q + 1)-hyperbolic σ -quadric of PG(3, q n ) with vertex points R and L, then n is even and has canonical equation , where x σ +1 = b has exactly q + 1 solutions. Moreover, | | = q 2n + (q + 1)q n + 1 and contains exactly q + 1 lines through R and exactly q + 1 lines through L.
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, 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., π : x 1 = 0, we get that the set ∩ π is a σ -conic of the plane π with associated matrix of rank 3. Hence, it is a Kestenband σ -conic of π . It follows that is a cone with vertex the point R projecting a Kestenband σ -conic in a plane not through R.
In this section, a σ -quadric of PG(3, q n ) will have equation X t AX σ = 0 with rk(A) ≤ 2.

-quadrics of rank 2
Since rk(A) = 2, it follows that the left and right radicals are two lines r and of PG(3, q n ). We distinguish three cases. 1) r ∩ = ∅. We may assume w.l.o.g. that r : x 3 = x 4 = 0 and : x 1 = x 2 = 0. Then: that is the set of points of PG(3, q n ) of intersection of corresponding planes under a collineation : P r −→ P , where The set contains the q n + 1 lines of a pseudoregulus (see [30]) with transversal lines r and We call this set a σ -quadric of pseudoregulus type with skew vertex lines r and .
Note that if n = 2, σ 2 = 1 a pseudoregulus have been already introduced in [15] and σ -quadrics of pseudoregulus type with skew vertex lines have been introduced in [11] where they were called hyperbolic Q F -sets. Let : P r −→ P s be a collineation with accompanying automorphism σ : x → x q m , (m, n) = 1. Assuming that : (π 1,0 ) = π 1,0 , (π 0,1 ) = π 0,1 , (π 1,1 ) = π 1,1 , we have that (π a,b ) = π a σ ,b σ . Hence, the set Q of points of intersection of corresponding planes under is given by the points whose homogeneous coordinates are the solutions of the linear system where (a, b) ∈ PG(1, q n ). This system is equivalent to the linear system 4 ) belongs to Q if and only if the previous linear system, in the unknowns a and b, has non-trivial solutions, hence if and only if This can be seen as a canonical equation of a σ -quadric of pseudoregulus type with skew vertex lines. Let c ∈ F * q and let γ ∈ F * q n be such that γ x σ , γ y σ , x, y)} (x,y)∈PG(1,q n ) .
The set Q c is a maximum scattered linear set of rank 2n of PG(3, q n ) of pseudoregulus type with transversal lines r and . Hence, the set Q is the union of the skew lines r , and the q − 1 linear sets of pseudoregulus type Q c , c ∈ F * q n . The following holds: 2) r ∩ = {V } is a point. We may assume w.l.o.g. that r : x 3 = x 4 = 0, : x 2 = x 3 = 0. In this case, the σ -quadric is a cone with vertex the point V projecting a (degenerate or not) C m F -set in a plane not through V . Indeed, let π be a plane not through the point V and let R = r ∩ π , L = ∩ π . It follows that ∩ π is a set of points of π generated by a collineation between the pencils of lines of π with center the points R and L induced by the collineation between the pencil of planes P r and P that is associated to . 3) r = . We may assume w.l.o.g. that r = : x 3 = x 4 = 0. In this case, the σ -quadric is a cone with vertex the line r over a σ -quadric of a line skew with r . That is 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 F q -subpencil of planes through r .

-quadrics of rank 1
In this subsection, a σ -quadric will have equation X t AX σ = 0 with rk(A) = 1. Hence, dimV ⊥ = dim V = 3 so left and right radicals in PG(3, q n ) are planes. We distinguish two cases: We may assume that r : x 4 = 0 is the right radical and : x 1 = 0 is the left radical. Hence, : x 1 x σ 4 = 0, that is the union of two different planes.
We may assume r = : x 4 = 0 is both the left and right radical. Hence, : In this section, we summarize the results obtained on σ -quadrics. Proposition 6.1 Let : X t AX σ = 0 be a σ -quadric of PG(3, q n ), with |A| = 0. The set is one of the following: • a cone with vertex a line v projecting a σ -quadric of a line 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 σ -conic or a (possibly degenerate) C m F -set of a plane π , with V / ∈ π , • a degenerate either parabolic or hyperbolic σ -quadric with two collinear vertex points, • a non-degenerate either elliptic or parabolic or hyperbolic or (q + 1)-hyperbolic σ -quadric with two vertex points, • a non-degenerate σ -quadric with two skew vertex lines (i.e., a σ -quadric of pseudoregulus type).
Let σ : x → x q m , σ : x → x q m , (m, n) = (m , n) = 1. We will say that a σ -quadric and a σ -quadric of PG(3, q n ) are of the same type if they have the same name. Proof Since and are not cones, both and have two distinct vertices. We divide two cases. Case 1. The vertices of both and are points. We may assume that and have the same vertices R and L. Let (resp. ) be the σ -collineation (resp. σ -collineation) between the star of lines with center R and the star of planes with center L associated to (resp. ). Observe that is also a σ −1 -quadric with vertices L and R to which there is associated the σ −1 -collineation −1 . Let f be a collineation of PG(3, q n ) mapping into . Since the number of lines of through either R or L is at most q + 1, there exists plane π through the line RL s.t. the lines of π through R are mapped under onto planes through a line on L not contained in π . It follows that induces a collineation π between the pencil of lines through R and the pencil of lines through L. This gives that ∩ π is a (possibly degenerate) C m F -set of π . It follows that f ( ∩ π) is a (possibly degenerate) either C m F or C n−m F -set. Hence, σ = σ or σ = σ −1 . Case 2. The vertices of both and are two skew lines. We may assume that and have the same vertices r and . Every plane not through r neither through meets in a (possibly degenerate) C m F set. The assertion follows in the same way as in the previous case.
Opposite to the case of PG(2, q n ), nothing is known for the sets of absolute points of PG(d, q n ) of a nonlinear correlation, different from a polarity, of PG(d, q n ), d ≥ 3. We recall that took roughly 14 years and 10 different papers to B. Kestenband to classify all σ -conics of PG(2, q n ) with equation X t AX σ = 0, |A| = 0. In this section, will be a σ -quadric of PG(d, q n ) with equation X t AX σ = 0, |A| = 0. We determine the dimension of the maximum totally isotropic subspaces contained in . Proof Let S be a subspace with maximum dimension, say h, contained in . We may assume, w.l.o.g., that S has equations x h+2 = x h+3 = . . . 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 then the σ -quadric with equation A similar proof can be given to obtain the following, more general, result An arc in 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 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 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 σ , t σ +1 )} ∪ {(0, 0, 0, 1)}, where σ : x → x 2 m , (m, n) = 1. The (q + 1)-arcs of PG(3, q) have been classified by Segre for q odd, q ≥ 5 [34] and by Casse and Glynn for q even, q ≥ 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 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 O\{P}. An example of translation ovoid of Q + (3, q) is a non degenerate conic. Proof Let r : x 3 = x 4 = 0 and : x 1 = x 2 = 0 and let be the σ -quadric with vertex lines and r with equation b ∈ F q n , stabilizes P ∞ and acts sharply transitively on the points of O\{P ∞ }.

Degenerate -quadrics and Lüneburg spread of PG(3, 2 n ).
In 1965, H. Lüneburg proved that if q n = 2 2h+1 , h ≥ 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 PG(3, q n ) (see [32]). Let ∞ be a hyperplane of PG(4, q n ) and let Q + (3, q n ) be a hyperbolic quadric of ∞ . A set A of q 2n points of PG(4, q n )\ ∞ s.t. the line joining any two of them is disjoint from Q + (3, q n ) is called an affine set of PG(4, q n )\ ∞ . In what follows, we will denote by ⊥ the polarity of Q + (5, q n ). In [29] and also in [31] the following result has been proved. If S is a spread of PG(3, q n ) and is a line of S, then we will denote by A (S) the affine set arising from S with respect to . If S is a symplectic spread, then A (S) is a set of q 2n points of an affine space PG(3, q n )\π ∞ such that the line joining any two of them is disjoint from a given non-degenerate conic C of π ∞ . 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: The affine set A of the Lüneburg spread of 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 A on π ∞ are the complement of a regular hyperoval.
With the following theorem, we observe that the affine set of a Lüneburg spread of PG (3, 2 2h+1 ) is the affine part of a degenerate elliptic σ -quadric of PG(3, 2 2h+1 ) with two vertex points. Proof Let be a degenerate parabolic σ -quadric with collinear vertex points R = (0, 0, 1, 0) and L = (0, 1, 0, 0) with equation It follows that the plane π RL has equation x 1 = 0. The set A = K\π RL is given by A = {(1, x, x σ + y σ +1 , y) : x, y ∈ F q n }. Arguing as in Proposition 5.2 in [8], we obtain that the set of directions determined by A into the plane π RL covers all the points of π RL except to the points of a hyperoval C given by C = {(0, x, x σ −2 , 1) : x ∈ F * q n }. Note that if q = 2 2h+1 and σ : x → x 2 h , then the hyperoval C is a hyperconic. The assertion follows (see Sect. 8.2). and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.