Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 38 (1997), No. 2, 289-298.

Quadratic Embeddings

Hans Havlicek Corrado Zanella

Abteilung für Lineare Algebra und Geometrie, Technische Universität, Wiedner Hauptstraße 8–10, A-1040 Wien, Austria Dip. di Matematica Pura ed Applicata, Università di Padova via Belzoni 7, I-35131 Padova, Italy

Abstract. The quadratic Veronese embedding ρ maps the point set P of PG(n, F) into the point set of PG(\binom{n+2}{2} − 1, F) (F a commutative field) and has the following well-known property: If M ⊂ P, then the intersection of all quadrics containing M is the inverse image of the linear closure of M^{ρ}. In other words, ρ transforms the closure from quadratic into linear. In this paper we use this property to define “quadratic embeddings”. We shall prove that if ν is a quadratic embedding of PG(n, F) into PG(n′, F′) (F a commutative field), then ρ^{−1}ν is dimension-preserving. Moreover, up to some exceptional cases, there is an injective homomorphism of F into F′. An additional regularity property for quadratic embeddings allows us to give a geometric characterization of the quadratic Veronese embedding.

1. Introduction

The aim of this paper is to examine geometric properties of a quadratic embedding, i.e. a mapping between projective spaces sharing some properties of the classical quadratic Veronese embedding. We follow an approach that has been used in discussing embeddings of Grassmann spaces (cf. [6] and [16]) and product spaces (cf. [17]). See also [8, chapter 25] for combinatorial characterizations of Veronese varieties over finite fields.

Let F be a commutative field and (P, L) := PG(n, F). Write

Φ := {S ⊂ P| S is a quadric of PG(n, F)} ∪ {P}.

If M ⊂ P, then the quadratic closure of M is

\overline{\overline{M}} := ⋂_{M⊂S, S∈Φ} S.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 1997 Heldermann Verlag

We call M a closed set if M = \overline{\overline{M}}. The linear closure of a set M of points will be denoted by \overline{M}. Each hyperplane of PG(n, F) is a quadric, namely a repeated hyperplane. Hence M ⊂ \overline{\overline{M}} ⊂ \overline{M}.

Definition 1. Let (P, L) := PG(n, F) and (P′, L′) := PG(n′, F′), where the field F is commutative. A mapping ν : P → P′ is a quadratic embedding if

\overline{\overline{M}} = (\overline{M^{ν}})^{ν^{−1}} for all M ⊂ P,  (1)

and

\overline{im ν} = P′.  (2)

We give some examples of quadratic embeddings:

Example 1. The classical quadratic Veronese embedding ρ is defined in the case F′ = F, n′ = \binom{n+2}{2} − 1, by

F (x_{0}, …, x_{n}) ⟼^{ρ} F (y_{ij})_{0≤i≤j≤n}, with y_{ij} := x_{i}x_{j}.

There are many equivalent definitions. Cf., e.g., [5], [7], [9].

Example 2. Let n′ = \binom{n+2}{2} − 1. If α : F → F′ is an injective homomorphism, then α induces a canonical embedding ϵ of PG(n′, F) into PG(n′, F′). The mapping ρϵ turns out to be a quadratic embedding. Since there are examples of fields admitting an injective, but not surjective homomorphism α : F → F, there exist quadratic embeddings different from the classical one, even if we demand that F and F′ are isomorphic.

Example 3. If n = 1 then M ⊂ P is closed if, and only if, |M| ≤ 2 or M = P. Thus a mapping ν : P → P′ is a quadratic embedding if, and only if, n′ = 2, ν is injective and im ν is an arc.

Example 4. If PG(n, F) = PG(2, 2) and F is a frame in PG(5, F′) then any injection ν : P → P′ such that im ν = F is a quadratic embedding. This is immediate from the fact that each subset M of P is closed, unless |M| = 6.

2. Properties of quadratic embeddings

In this section ν is a quadratic embedding of PG(n, F) = (P, L) (n ≥ 1) into PG(n′, F′) = (P′, L′).

Proposition 2.1. Let K_{1} and K_{2} be two distinct closed sets in PG(n, F). Then \overline{K_{1}^{ν}} ≠ \overline{K_{2}^{ν}}. Consequently, the mapping ν is injective and satisfies

\overline{\overline{M}}^{ν} = \overline{M^{ν}} ∩ im ν for all M ⊂ P.  (3)

If U′ ⊂ P′ is a subspace, then U^{′ν^{−1}} ⊂ P is a closed set.

Proof. By the definition of a quadratic embedding,

(\overline{K_{1}^{ν}})^{ν^{−1}} = \overline{\overline{K_{1}}} = K_{1} ≠ K_{2} = \overline{\overline{K_{2}}} = (\overline{K_{2}^{ν}})^{ν^{−1}},

so that \overline{K_{1}^{ν}} ≠ \overline{K_{2}^{ν}}. Moreover, ν is injective, since any subset of P with a single element is closed. Hence (3) is true. Finally, let M := U^{′ν^{−1}}. Then

\overline{\overline{M}} = (\overline{M^{ν}})^{ν^{−1}} ⊂ U^{′ν^{−1}} = M.  □

Theorem 1. If ν is a quadratic embedding of PG(n, F) into PG(n′, F′), then n′ = \binom{n+2}{2}−1.

Proof. Define δ(t) := \binom{t+2}{2}, t ∈ ℕ. Let \{e_{0}, …, e_{n}\} be a basis of F^{n+1} and

X := {F (e_{i} + e_{j})|0 ≤ i < j ≤ n} ∪ {F e_{i}|i = 0, …, n}.

Since |X | = δ(n) and \overline{\overline{X}} = P, we have im ν ⊂ \overline{X^{ν}}, hence

n′ ≤ δ(n) − 1.  (4)

We now prove that in (4) the equality holds. We give a definition, by recursion on d = 0, …, n, of distinct closed sets in PG(n, F), say K_{δ(d−1)}, K_{δ(d−1)+1}, …, K_{δ(d)−1}, such that

K_{δ(d−1)} ⊂ K_{δ(d−1)+1} ⊂ … ⊂ K_{δ(d)−1},

with ^{d}U := K_{δ(d)−1} being a d-subspace of PG(n, F). For d = 0, choose a point Q and let K_{0} = ^{0}U := {Q}. Now let d > 0 and K_{δ(d−1)−1} = ^{d−1}U. Take a d-subspace ^{d}U containing d−1U and a basis B = {P_{0}, …, P_{d}} of ^{d}U such that ^{d−1}U ∩ B = ∅. Since the union of two subspaces of PG(n, F) is a closed set, we can define

K_{δ(d−1)+i} := ^{d−1}U ∪ \overline{\{P_{0}, …, P_{i}\}}, i ∈ \{0, …, d\}.

By Proposition 2.1,

∅ ⊂ \overline{K_{0}^{ν}} ⊂ \overline{K_{1}^{ν}} ⊂ … ⊂ K_{δ(n)−1}^{ν}

is a chain of distinct subspaces of PG(n′, F′). □

Proposition 2.2. If M ⊂ P, then dim(\overline{M^{ν}}) is equal to the largest i ∈ ℕ, such that there exists a chain

∅ ⊂ K_{0} ⊂ K_{1} ⊂ … ⊂ K_{i} = \overline{\overline{M}},  (5)

consisting of i + 2 distinct closed subsets of \overline{\overline{M}}. Consequently, dim(\overline{M^{ν}}) = dim(\overline{M^{ρ}}), where ρ denotes the quadratic Veronese embedding.

Proof. By Proposition 2.1, the subspaces \overline{K_{0}^{ν}}, \overline{K_{1}^{ν}}, …, \overline{K_{i}^{ν}} are distinct and

\overline{K_{i}^{ν}} = \overline{\overline{M^{ν}} ∩ im ν} = \overline{M^{ν}},

whence dim(\overline{M^{ν}}) ≥ i.

Now assume dim(\overline{M^{ν}}) > i. Then there exists an integer j, 0 ≤ j < i, such that
dim(\overline{K_{j+1}^{ν}}) ≠ dim(\overline{K_{j}^{ν}}) + 1.

Let P ∈ K_{j+1}^{ν} \ \overline{K_{j}^{ν}}. Then K := (\overline{\{P \} ∪ K_{j}^{ν}})^{ν^{−1}} is a closed set (cf. Proposition 2.1), and K_{j} ⊂ K ⊂ K_{j+1}, K ≠ K_{j}. The maximality of the chain (5) implies K = K_{j+1}. Therefore

dim(\overline{K_{j+1}^{ν}}) = dim(\overline{\{P \} ∪ K_{j}^{ν}}) = dim(\overline{K_{j}^{ν}}) + 1,

a contradiction. □

Proposition 2.3. If T is a hyperplane of PG(n, F) and M ⊂ P \ T , then

dim(\overline{(T ∪ M)^{ν}}) = \binom{n + 1}{2} + dim(\overline{M}).  (6)

Proof. By Theorem 1, dim(\overline{T^{ν}}) = \binom{n+1}{2} − 1. If B = \{P_{0}, …, P_{t}\} ⊂ M is a basis of \overline{M}, then the closed sets

K_{i} := T ∪ \overline{\{P_{0}, …, P_{i}\}}, i ∈ \{0, …, t\},

form a saturated chain

T ⊂ K_{0} ⊂ … ⊂ K_{t}.

Now the assertion is a consequence of Theorem 1 and Proposition 2.2. □

Proposition 2.4. Let T be a hyperplane of PG(n, F). If \overline{T^{ν}} and E′ are complementary subspaces of PG(n′, F′), then the mapping

ι : P \ T ⟶ E′ : A ⟼ \overline{(T ∪ \{A\})^{ν}} ∩ E′  (7)

has the following property:

dim(\overline{M}) = dim(\overline{M^{ι}}) for all M ⊂ P \ T .  (8)

Consequently, ι is preserving both collinearity and non-collinearity of points. So, the mapping ι is a (linear) embedding of the affine space P \ T into the projective space E′.

Proof. By (6), dim \overline{T^{ν}} = \binom{n+1}{2} − 1, whence dim E′ = n. Applying (6) again yields

dim(\overline{M^{ι}}) = dim(\overline{T^{ν} ∪ M^{ν}} ∩ E′) = dim(\overline{T^{ν} ∪ M^{ν}}) − (dim(\overline{T^{ν}}) + 1)
= dim(\overline{M}).  □

Proposition 2.5. Let |F | > 2 and n ≥ 2. If |F | ≠ 3 or n ≠ 2, then the embedding (7) can be extended to exactly one embedding β : P → E′.

Proof. The case n = 2 is dealt with in [12]. The case |F | > 3 is covered by [2, Theorem 3.5]. Thus only |F | = 3 and n > 2 remains open. For F′ being finite, the assertion follows from a result in [10, chapitre 2.3] (cf. also [11, théorème 1]), and by slight modifications, this carries over to an infinite F′.

On the other hand we sketch a direct proof for n > 2: Let P ∈ T . If g, h ∈ L and P ∈ g ∩ h, then the lines \overline{(g \ \{P \})^{ι}} and \overline{(h \ \{P \})^{ι}} are coplanar by Proposition 2.4. Since there exist three non coplanar lines through P, all lines of the kind \overline{(g \ \{P \})^{ι}}, with P ∈ g, share one point P′ ∈ E′. Then we define P^{β} := P′. By repeatedly using Proposition 2.4, we have that β is an embedding. The restriction of β to a plane A of P, not contained in T , is an extension of ι|(A \ T ), and thus we obtain the uniqueness of β. □

As a corollary, we have:

Theorem 2. Let |F | > 2 and n ≥ 2. If |F | ≠ 3 or n ≠ 2, then the existence of a quadratic embedding of PG(n, F) into PG(n′, F′) implies that the field F is isomorphic to a subfield of F′. □

Whenever for some fixed hyperplane T ⊂ P and an adequately chosen subspace E′ ⊂ P′ the mapping (7) is uniquely extendable to an embedding β : P → E′, then T gives rise to an embedding

ν_{T} : P ⟶ P′/\overline{T^{ν}} : X ⟼ \{X^{β}\} ∨ \overline{T^{ν}};  (9)

here P′/\overline{T^{ν}} denotes the point set of the quotient space PG(n′, F′) modulo \overline{T^{ν}}. Moreover, we can associate with T the following hyperplane of PG(n′, F′):

\overline{T^{ν} ∪ T^{β}} =: H′_{T} .

Both definitions do not depend on the choice of E′. Since H′_{T} ∩ E′ ∩ im ι = ∅, we have

(H′_{T})^{ν^{−1}} = T .  (10)

Proposition 2.6. Let Σ be the collection of all hyperplanes of PG(n, F). If H′_{T} is well defined for all T ∈ Σ, then

ν̂ : Σ ⟶ Σ′ : T ⟼ H′_{T}  (11)

is an injective mapping. □

The previous results give sufficient conditions for the existence of the mapping ν̂.

3. Regular quadratic embeddings

In the following we shall assume that ν is a quadratic embedding of PG(n, F), n ≥ 1, into PG(n′, F′), n′ = \binom{n+2}{2} − 1.

Definition 2. A quadratic embedding ν is called (P, ℓ)-regular if there exists an incident point-line pair (P, ℓ) of PG(n, F) such that the plane arc ℓ^{ν} has a unique unisecant line which is running through P^{ν} and contained in the plane \overline{ℓ^{ν}}. If ν is (P, ℓ)-regular for all incident pairs (P, ℓ), then ν is said to be a regular quadratic embedding.

Proposition 3.1. Suppose that n ≥ 2 and that ν is (P, ℓ)-regular. Then ν is regular.

Proof. By n ≥ 2, there exists a hyperplane T of PG(n, F) such that P ∈ T , ℓ ⊄ T . Define an embedding ι : P \ T → E′ according to (7). The (P, ℓ)-regularity of ν implies that

|\overline{(ℓ \ \{P \})^{ι}} \ (ℓ \ \{P \})^{ι}| = 1.  (12)

If the settings of Proposition 2.5 are true, then ι extends to an embedding β : P → E′ with ℓ^{β} = \overline{(ℓ \ \{P \})^{ι}} by (12). Hence β is a collineation.

Otherwise |F | =: p ∈ {2, 3} so that |F′| = p by (12). If X ∈ P \T , then there is a certain number of lines through X and on each such line there are p points of P \ T . In E′ the same number of lines is running through X^{ι} and, by Proposition 2.4, there are p points of im ι on each such line. This in turn means that on each line in E′ through a point of E′ \ im ι there is either no point of im ι or no other point of E′ \ im ι, whence E′ \ im ι is a subspace. More precisely, T′ := E′ \ im ι is a hyperplane of E′. Two distinct lines of the affine space P \ T are parallel if, and only if, they are disjoint and coplanar. By Proposition 2.4 these properties carry over to the ι-images of these lines, whence ι is an affinity of P \ T onto E′ \ T′. (This is trivial when p = 3.) Thus ι is also extendable to a collineation β : P → E′ if Proposition 2.5 cannot be applied.

Next choose any point X_{1} ∈ T and any line ℓ_{1} ⊄ T , X_{1} ∈ ℓ_{1}. Then (\{X_{1}^{β}\} ∨ \overline{T^{ν}}) ∩ \overline{ℓ_{1}^{ν}} is the only unisecant of ℓ_{1}^{ν} at X_{1}^{ν} within the plane \overline{ℓ_{1}^{ν}}, since |\overline{(ℓ_{1} \ \{X_{1}\})^{ι}} \ (ℓ_{1} \ \{X_{1}\})^{ι}| = 1. Hence ν is (X_{1}, ℓ_{1})-regular. Repeatedly using this last idea yields that ν is regular. □

As an immediate consequence of the proof of Proposition 3.1 we have

Proposition 3.2. Let ν be a regular quadratic embedding and n ≥ 2. Choose any hyperplane T ⊂ P. Then the embedding ι : P \ T → E′, defined according to (7), is extendable to a unique collineation β : P → E′. Consequently, F and F′ are isomorphic fields, and ν_{T} : P → P′/\overline{T^{ν}} (cf. (9)) is a collineation. □

If n = 1, then the (P, ℓ)-regularity of ν implies

|F | = |ℓ \ {P }| = |ℓ^{ν} \ {P^{ν}}| = |F′|.

This does not imply, however, that ν is regular. If n = 1 and ν is regular, then im ν obviously is an oval but not necessarily a conic. Cf. [3, 4, 14] for topological conditions that force an oval to be a conic.

These results can be improved if we assume that |F | =: q is finite. Then n = 1 and ν being (P, ℓ)-regular yield |F | = |F′| = q so that im ν is a (q + 1)-arc in PG(2, F′) ≅ PG(2, q). Hence im ν is an oval, which in turn shows that ν is regular. Moreover, by Segre’s theorem, im ν is a (regular) conic if q is odd; the last result is also true when q ∈ {2, 4}.

The case n = 1 is excluded from our further discussions. Hence we may assume without loss of generality that F = F′, by virtue of Proposition 3.2.

The following result will be used in order to characterize the ν-images of lines.

Lemma 1. Let P_{0} and P_{2} be two distinct points of PG(2, F) and let σ be a collineation of PG(2, F) taking P_{0} to P_{2}, but not fixing the line \overline{P_{0}P_{2}}. Then

C := {X|{X} = x ∩ x^{σ}, x is a line through P_{0}}

is containing three distinct collinear points if, and only if, σ is a non-projective collineation.

Proof. If σ is projective, then C is a regular conic, whence it does not contain three distinct collinear points.

If σ is not projective, then let α ∈ Aut(F) be the companion automorphism of σ. Set

\{P_{1}\} := (\overline{P_{0}P_{2}})^{σ^{−1}} ∩ (\overline{P_{0}P_{2}})^{σ}.

Choose some line e running through P_{0} but not containing P_{1} or P_{2} and define {E} := e∩e^{σ}. Then (P_{0}, P_{1}, P_{2}, E) is an ordered quadrangle; we may assume that this is the standard frame of reference. A straightforward calculation yields

C = {F (u_{0}u_{0}^{α}, u_{0}u_{1}^{α}, u_{1}u_{1}^{α})|(0, 0) ≠ (u_{0}, u_{1}) ∈ F^{2}}.

By α ≠ id_{F}, there exists an element c ∈ F with c ≠ c^{α}. Define v ∈ F via c = v^{αα}c^{α}. Thus v ≠ 0, 1 and F (1, 1, 1), F (1, v^{α}, vv^{α}) are distinct points of C. With

w := \frac{1 + vv^{α}c}{1 + v^{α}c}

we obtain

\frac{1}{1 + c}(1, 1, 1) + \frac{c}{1 + c}(1, v^{α}, vv^{α}) = (1, w^{α}, ww^{α}),

whence C is containing three distinct collinear points. □

Proposition 3.3. If g is a line of PG(n, F), n ≥ 2, and ν is a regular quadratic embedding, then g^{ν} is a regular conic.

Proof. Choose hyperplanes T , Ǔ ⊂ P such that g ∩ T ∩ Ǔ = ∅. Set {T } := g ∩ T and define a collineation ν_{T} according to (9). Write

L′_{T} := \{x′ ∈ L′|T^{ν} ∈ x′ ⊂ \overline{g^{ν}}\}

and

π_{T} : L′_{T} ⟶ g^{ν_{T}} : x′ ⟼ x′ ∨ \overline{T^{ν}}.

This π_{T} is a projectivity from a pencil of lines onto a pencil of subspaces. Replacing T by Ǔ gives a point U and a projectivity π_{U}. Since ν_{T}^{−1}νǓ is a collineation of quotient spaces,

π_{T} ν_{T}^{−1}νǓπ_{U}^{−1} : L′_{T} ⟶ L′_{U}

is extendable to a collineation, say σ, of the plane \overline{g^{ν}} onto itself. We have

\begin{matrix}\overline{g^{ν}} ∩ T ν̂ ⟼^{σ} & \overline{T^{ν}U^{ν}} ≠ \overline{g^{ν}} ∩ T ν̂, \\ \overline{T^{ν}U^{ν}} ⟼^{σ} & \overline{g^{ν}} ∩ Ǔ ν̂, \\ \overline{T^{ν}X^{ν}} ⟼^{σ} & \overline{U^{ν}X^{ν}} (X ∈ g \ \{T, U\}) \end{matrix}

and

g^{ν} = {X′|{X′} = x′ ∩ x^{′σ}, x′ ∈ L′_{T} }.

Since any two distinct points of g form a closed set, no three points of g^{ν} are collinear. We read off from Lemma 1 that σ is projective, whence g^{ν} is a regular conic. □

We remark that ν_{T}^{−1}νǓ is a projective collineation of quotient spaces.

Proposition 3.4. Let (P_{0}, P_{1}, …, P_{n}, E) be an ordered frame of PG(n, F), n ≥ 2, and let ν be a regular quadratic embedding. Write Q′_{ii} := P_{i}^{ν}, E′ := E^{ν}, and Q′_{ij} for the common point of the tangent lines of the conic (\overline{P_{i}P_{j}})^{ν} at P_{i}^{ν} and P_{j}^{ν}, i, j ∈ \{0, 1, …, n\}, i ≠ j. Then

{Q′_{ij}|0 ≤ i ≤ j ≤ n} ∪ {E′}  (13)

is a frame of PG(n′, F).

Proof. For any i, j ∈\{0, 1, …, n\}, i < j, take a P_{ij} ∈ \overline{P_{i}P_{j}}\\{P_{i}, P_{j}\}. Then \{Q′_{00}, Q′_{11}, …, Q′_{nn}\} ∪ {P_{ij}^{ν}|0 ≤ i < j ≤ n} is a basis of PG(n′, F) by Theorem 1. Since Q′_{ii}, Q′_{jj}, P_{ij}^{ν}, Q′_{ij} is a plane quadrangle, the exchange lemma yields that

B′ := {Q′_{ij}|0 ≤ i ≤ j ≤ n}

is a basis of PG(n′, F).

Define hyperplanes

T_{i} := \overline{\{P_{k}|k ∈ \{0, 1, …, n\} \ \{i\}\}} ⊂ P,

and X′_{i} := T ν̂ ; i (cf. (11)) for i ∈ {0, 1, …, n}. Obviously Q′_{jk} ∈ X′_{i} for all j, k ∈ {0, 1, …, n} \

{i}. Moreover, if j ∈ {0, 1, …, n} \ {i}, then

\overline{\{Q′_{ii}, Q′_{ij}, Q′_{jj}\}} ∩ X′_{i}

is the tangent line of the conic (\overline{P_{i}P_{j}})^{ν} at P_{j}^{ν} = Q′_{jj}, so that Q′_{ij} ∈ X′_{i}. We infer that

X′_{i} = \overline{B′ \ \{Q′_{ii}\}}.

Now E ∉ T_{i} implies E′ ∉ X′_{i}. Finally, T_{i} ∪ T_{j} is a closed set not containing E. Hence

E′ ∉ \overline{(T_{i} ∪ T_{j})^{ν}} = \overline{B′ \ \{Q′_{ij}\}}.

This completes the proof. □

Theorem 3. If ν is a regular quadratic embedding of PG(n, F) into PG(n′, F), n ≥ 2, n′ = \binom{n+2}{2}−1 and ρ denotes the quadratic Veronese embedding, then there exists a collineation κ of PG(n′, F) such that ν = ρκ.

Proof. We adopt the notation of Proposition 3.4. The coordinates with respect to (P_{0}, P_{1}, …, P_{n}, E) of a point X ∈ P are written as F (x_{0}, x_{1}, …, x_{n}), and the coordinates of X^{ν} with respect to (Q′_{00}, Q′_{01}, …, Q′_{nn}, E′) (cf. (13)) are denoted by F (y_{00}, y_{01}, …, y_{nn}). In order to simplify notation we put y_{ij} := y_{ji} for i > j.

Choose an index i ∈ {0, 1, …, n} and set

E′_{i} := \overline{\{Q′_{i0}, Q′_{i1}, …, Q′_{in}\}}.

Hence E′_{i} is a complement of \overline{T_{i}^{ν}} (cf. the proof of Proposition 3.4) and, by Proposition 3.2, we obtain a collineation β_{i} : P → E′_{i} with P_{j} ↦ Q′_{ij} (j ∈ {0, 1, …, n}) and E ↦ E′_{i}, where \{E′_{i}\} := (\{E′\} ∨ \overline{T_{i}^{ν}}) ∩ E′_{i}. So, by taking (Q′_{i0}, Q′_{i1}, …, Q′_{in}, E′_{i}) as frame of reference in E′, we obtain that X^{β_{i}} has coordinates

F (x_{0}^{α_{i}}, x_{1}^{α_{i}}, …, x_{n}^{α_{i}})

with α_{i} ∈ Aut(F).

If j ∈ {0, 1, …, n}, then ν_{T_{i}}^{−1}ν_{T_{j}} is a projective collineation, as has been remarked after Proposition 3.3. Hence also β_{i}^{−1}β_{j} is projective. Thus β_{i} and β_{j} belong to the same automorphism α := α_{i} ∈ Aut(F).

Now we compare the coordinates of X, X^{ν}, X^{β_{i}}: If x_{i} ≠ 0, then \{X^{β_{i}}\} = (\{X^{ν}\}∨\overline{T_{i}^{ν}})∩E′_{i}. Hence there exists an element c_{i} ∈ F \ {0} such that

y_{i0} = c_{i}x_{0}^{α}, y_{i1} = c_{i}x_{1}^{α}, …, y_{in} = c_{i}x_{n}^{α}.

If, moreover, x_{j} ≠ 0, j ∈ {0, 1, …, n} \ {i}, then

c_{i} = \frac{y_{ij}}{x_{j}^{α}} , c_{j} = \frac{y_{ji}}{x_{i}^{α}}

whence, by y_{ij} = y_{ji},

\frac{c_{i}}{c_{j}} = \frac{x_{i}^{α}}{x_{j}^{α}} .

If x_{i} = 0, then y_{i0} = y_{i1} = … = y_{in} = 0. Thus we have

F (y_{00}, y_{01}, …, y_{nn}) = F (x_{0}^{α}x_{0}^{α}, x_{0}^{α}x_{1}^{α}, …, x_{n}^{α}x_{n}^{α}).

Now letting κ be that collineation of PG(n′, F) which transforms each coordinate under α completes the proof. □

References

[1] Bichara, A.; Korchmáros, G.: n^{2}-sets in a projective plane which determine exactly n^{2}+n lines. J. Geom. 15 (1980), 175–181.

[2] Brezuleanu, A.; Radulescu, D. C.: About full or injective lineations. J. Geom. 23 (1984), 45–60.

[3] Buchanan, T.: Ovale und Kegelschnitte in der komplexen projektiven Ebene. Math. Phys. Sem. Ber. 26 (1979), 244–260.

[4] Buchanan, T.; Hähl, H.; Löwen, R.: Topologische Ovale. Geom. Dedicata 9 (1980), 401–424.

[5] Burau, W.: Mehrdimensionale projektive und höhere Geometrie. Dt. Verlag d. Wissenschaften, Berlin 1961.

[6] Havlicek, H.: Zur Theorie linearer Abbildungen I, II . J. Geom. 16 (1981), 152–180.

[7] Herzer, A.: Die Schmieghyperebenen an die Veronese-Mannigfaltigkeit bei beliebiger Charakteristik. J. Geom. 18 (1982), 140–154.

[8] Hirschfeld, J. W. P.; Thas, J. A.: General Galois Geometries. Oxford University Press 1991.

[9] Karzel, H.: Über einen Fundamentalsatz der synthetischen algebraischen Geometrie von W. Burau und H. Timmermann. J. Geom. 28 (1987), 86–101.

[10] Limbos, M.: Plongements et Arcs Projectifs. Thèse Université Libre de Bruxelles, 1980/81.

[11] Limbos, M.: Plongements de PG(n, q) et AG(n, q) dans PG(m, q′), m < n. C. R. Rep. Acad. Sci. Canada 4 (1982), 65–68.

[12] Rigby, J. F.: Affine subplanes of finite projective planes. Canad. J. Math. 17 (1965), 977–1009.

[13] Schröder, E. M.: Fundamentalsätze der metrischen Geometrie. J. Geom. 27 (1986), 36– 59.

[14] Strambach, K.: Projektive Ebenen und Kegelschnitte. Arch. Math. 64 (1995), 170–184.

[15] Timmermann, H.: Descrizioni geometriche sintetiche di geometrie proiettive con caratteristica p > 0. Ann. Mat. Pura Appl. 114 (1977), 121–138.

[16] Wells Jr., A. L.: Universal projective embeddings of the Grassmannian, half spinor, and dual orthogonal geometries. Quart. J. Math. Oxford 34 (1983), 375–386.

[17] Zanella, C.: Universal properties of the Corrado Segre embedding. Bull. Belg. Math. Soc. 3 (1996), 65–79.

Received December 8, 1995; revised version December 10, 1996