Veronesean representations of projective planes over
quadratic alternative division algebras
O. Krauss
J. Schillewaert
∗
H. Van Maldeghem
Abstract
We characterize the Veronesean representations of all the Moufang projective
planes defined over a quadratic alternative division algebra, including the new
type of Veroneseans for the inseparable case in characteristic 2, by means of the
Mazzocca-Melone axioms, which essentially express three elementary properties of
these representations as algebraic and differential varieties over the real field.
1
Introduction
The Veronesean representation of the Moufang projective plane P(O) related to any
Cayley-Dickson division algebra O is the geometry of an important real form of a group
of exceptional type E6 . One of its nice features is that it provides a homogeneous description of these projective planes, which otherwise must be defined via the projective
closures of the corresponding affine planes (and this places a certain line in a special role).
The subplanes of P(O) related to its subalgebras have induced representations and similar properties. Three of these properties have been used by Mazzocca & Melone [8] to
characterize the finite quadric Veroneseans. More generally, the objects satisfying these
axioms are called Veronesean caps. Despite many special cases of Veronesean caps have
been classified (among which all finite ones), until to date, there was no full classification.
In the present paper, we present such a classification, and we find besides the “classical”
Veronesean caps, also a class related to inseparable field extensions in characteristic 2
(which can also be viewed as quadratic algebras).
The results of the present paper, together with the Main Result of [11], provide a common
characterization of the varieties related to the second row of the Freudenthal-Tits Magic
Square, both the split and nonsplit version. This corresponds to a characterization of the
class of all “projective planes” defined over quadratic alternative composition algebras,
∗
Research supported by Marie Curie IEF grant GELATI (EC grant nr 328178)
1
Remarkably, in the nonsplit case, additionally inseparable cases occur, while in the split
case, additionally only some other sporadic varieties occur such as those related to the halfspin D5 -geometry, to the line Grassmannian of projective 4-space and the Segre varieties
related to projective spaces of dimension at most 2.
The first three cells of the second row of the Freudenthal-Tits Magic Square admit a
generalization to higher dimensions (Veronesean representations of projective spaces over
quadratic alternative division algebras). These can also be characterized using a similar
approach, but we will do this elsewhere. A big part of this is also contained in the first
author’s thesis [7]. The split case has recently been treated in [12].
Some notation. In this paper, the n-dimensional projective space over the skew field
K is denoted by Pn (K) (it arises from a right (n + 1)-dimensional vector space). The
projective space arising from a right vector space V is denoted by P(V ). We will use
a slightly different notation for the projective plane defined over a quadratic alternative
division ring A, which we will by P2 (A). This should clearly distinguish the geometries
we represent (the P(A)’s) and those in which we represent these (the P(V )’s).
Concerning projective planes, we will use the following terminology. A collineation is a
bijection between two projective planes preserving the line sets in both ways (it is allowed
that the two planes are identical and then we have a permutation of the points set inducing
a permutation of the line set). A translation line in a projective plane is a line L with
the property that the collineations fixing L pointwise and not fixing any point off L act
transitively on the set of points off L. A projective plane containing a translation line is
called a translation plane. A Moufang plane, or a plane satisfying the Moufang condition,
is a projective plane in which every line is a translation line. The projective planes P(A)
with A an quadratic alternative division ring are all Moufang planes. Conversely, every
Moufang plane is isomorphic to such a P(A).
Concerning projective spaces P(V ), with V a right vector space over some skew field K,
we will denoted the subspace spanned by a (point) set S with hSi. We usually view
subspaces as sets of points. Also, we will need to consider projections. These are maps
with a projection center U and an image W , where U and W are complementary subspaces,
i.e., U ∩ W = ∅ and hU, W i is the whole projective space. The projection from U to W
maps a point p ∈
/ U to the point hU, pi ∩ W . The projection from U onto W is not defined
on the points of U , and if we consider a subspace S intersecting U , then by the projection
of S we mean the subspace hU, Si ∩ W (so for subspaces, we do not project pointwise as
otherwise we always have to avoid the points of U ∩ S, which only makes the notation
cumbersome).
We end this introduction by mentioning that large parts of Section 6 are taken from
the PhD thesis of the first author, who obtained a partial characterization of V2 (K, A),
(mainly for the case of char K 6= 2), see below.
2
2
Definitions and Statement of the Main Results
An ovoid O in a possibly infinite-dimensional projective space Σ is a set of points of Σ
such that no line of Σ intersects O more than 2 points, and for every point x ∈ O, there is
a unique hyperplane π through x intersecting O in only x and containing all lines through
x that meet O in only x. The hyperplane π is called the tangent hyperplane at x to O
and denoted Tx (O). The dimension of the ovoid is equal to −1 + dim Σ (and this can be
infinite).
Let V be possibly infinite-dimensional right vector space over some skew field K, and let
P(V ) be the corresponding projective space. Let X be a spanning point set of P(V ) and
let Ξ be a collection of subspaces of P(V ), which we shall refer to as the elliptic spaces of
X, such that, for any ξ ∈ Ξ, the intersection ξ ∩ X is an ovoid X(ξ) in ξ of dimension
at least 1 (and then, for x ∈ X(ξ), we sometimes denote Tx (X(ξ)) simply by Tx (ξ)). We
call (X, Ξ), or briefly X, a Veronesean cap if the following properties (V1), (V2) and (V3)
hold.
(V1) Any two points x and y of X lie in a unique element of Ξ, denoted by [x, y].
(V2) If ξ1 , ξ2 ∈ Ξ, with ξ1 6= ξ2 , then ξ1 ∩ ξ2 ⊂ X.
(V3) For every x ∈ X and every triple of distinct points y1 , y2 , y3 ∈ X \ {x}, we have
Tx ([x, y3 ]) ⊆ hTx ([x, y1 ]), Tx ([x, y2 ])i.
If V is finite-dimensional we say that (X, Ξ) is finite-dimensional. Ovoids which are the
intersection of an elliptic space with X will be called X-ovoids. If all X-ovoids have the
same dimension d, then we say that X has subdimension d. We will prove in Lemma 4.1
that every Veronesan cap has a subdimension. Also, we will show in Lemma 4.9 that
(X, Ξ) is completely determined by X alone, and so we sometimes speak of the Veronesean
cap X.
We present a class of examples of Veronesean caps.
Let A be a quadratic alternative division algebra over the field K, where we assume K ⊆ A.
This means that A is a division ring, the multiplication is alternative (thus (ab)b = ab2
and a(ab) = a2 b, for all a, b ∈ A), K is a subfield of the center of A and every element
x ∈ A satisfies a quadratic equation with coefficients in K. Recall that one of the following
situations holds.
• A = K (and dimK A = 1);
• A is a quadratic Galois extension of K (and dimK A = 2);
• A is a quaternion division algebra with center K (and dimK A = 4);
• A is a Cayley-Dickson division algebra with center K (and dimK A = 8);
3
• A is a purely inseparable extension of K in characteristic 2 containing all squares of
K (and either dimK A is infinite or dimK A = 2` , with ` ≥ 2).
In each case there exists a unique involution σ on A, which is an anti-automorphism,
which is the identity on K and for which both a + aσ and aaσ belong to K (and we call
σ the standard involution). In fact, every a ∈ A then satisfies the quadratic equation
x2 − (a + aσ )x + aσ a = 0.
Then we can define the following 1-spaces of the vector space V := K⊕K⊕K⊕A⊕A⊕A.
• K(1, 0, 0; 0, 0, 0);
• K(xxσ , 1, 0; 0, 0, x), x ∈ A;
• K(xxσ , yy σ , 1; y, xσ , xy σ ), x, y ∈ A.
The corresponding set X of points of P(V ) is a Veronesean cap of subdimension dim A, see
Theorem 3 below, which we denote by V2 (K, A). In fact, if A is not a purely inseparable
extension of K, then X is the standard Veronesean embedding of the projective plane
P2 (A) over A (or, in other words, coordinatized by A): either the quadratic Veronesean
(A = K), or the Hermitian Veronesean (A is a quadratic Galois extension of K), or the
quaternion Veronesean (A is a quaternion division algebra with center K), or the octonion
Veronesean (A is a Cayley-Dickson division algebra with center K). If A is an inseparable
extension of K, then this is a new type of Veronesean, which is, as far as we know, not yet
studied in the literature, and which we shall call an inseparable Veronesean. In all cases,
the points of P2 (A) are in bijective correspondence with the elements of X, and the lines
of P2 (A) correspond with the X-ovoids, which are all quadrics of Witt index 1.
If A is associative, then one can define the Veronesean Vn (K, A) of Pn (A) over K, n ≥ 2,
explaining the index 2 in the notation. We will not be concerned with Vn (K, A) for
n 6= 2, but the present paper lays the foundation to characterize Vn (K, A) with n finite
in a similar way, using the techniques of [3]. This will be done elsewhere, as we do not
consider it as essential as the case of V2 (K, A).
We can now state our Main Result.
Main Result. Every Veronesean cap is projectively equivalent to V2 (K, A), for some
commutative field K and some quadratic alternative division algebra A over K. In particular, the subdimension is either infinite or a power of 2, and if the characteristic of the
underlying field is not 2, then the subdimensions can only take the values 1, 2, 4, 8.
Some special cases of the Main Result are already known in the literature. We give a
historic overview.
Mazzocca & Melone [8] were the first to define Veronesean caps; they defined Veronesean
caps X of subdimension 1 over finite fields, where they assumed the X-ovoids to be conics
4
(and not just ovals) and they classified them in the case of odd characteristic (although
they erroneously thought they classified these also in characteristic 2). Later, Hirschfeld &
Thas [5] corrected the error in characteristic 2, and Thas & Van Maldeghem [15] weakened
the assumptions on the X-ovoids to ovals and classified. In the meantime, Cooperstein,
Thas & Van Maldeghem [3] introduced and classified Veroneseans caps of subdimension
2 over finite fields. Then, the last two present authors (JS+HVM) classified Veronsean
caps of subdimension 1 and 2 over arbitrary fields in [9, 10] by showing that these are
automatically the quadratic Veronesean varieties V2 (K, K) and the Hermitian Veronesean
varieties V2 (K, L) (with L a quadratic Galois extension of K), respectively. However,
the paper [9] contains an incompleteness, which we rectify in Section 4.4 (basically, the
inseparable Veroneseans with subdimension 2 were overlooked; the two last authors thank
the first author for noting this error). Finally, the first author, in his thesis [7], recently
classified Veronesean caps under some additional conditions excluding characteristic 2, but
including the quaternion cases for which the center admits only two quadratic classes,
and the octonion case assuming, in the terminology of the present paper, the cap has
subdimension 8 (see below for precise definitions).
The main difficulty in handling the case of subdimension at least 3 is to deduce that
the underlying field K of the ambient projective space is contained in the center of the
algebra A (it is not so hard to construct the algebra; this follows from the fact that the
Veronesean cap X endowed with the X-ovoids, the geometry of which we shall denote by
P(X, Ξ), is a projective plane (see Proposition 4.5 below)). The crux of the argument
is then to construct a sub-Veronesean cap X 0 of subdimension 2 containing an arbitrary
sub-Veronesean cap X 00 of subdimension 1 and an additional point on the X-ovoid which
contains some arbitrarily chosen X 00 -ovoid. Notice that it is relatively standard to construct all sub-Veronesean caps of subdimension 1, but it is much harder to construct a
sub-Veronesean cap of subdimension 2. This is the heart of our proof. Moreover, in order
to fully include the inseparable case, we must deal with the possibility of the X-ovoids
having infinite dimension.
The entire proof of the Main Result is organized as follows. In Section 4, we show that
every Veronesean cap has a subdimension and that P(X, Ξ) is a Moufang projective plane.
We also review Veronesean caps of subdimensions 1 and 2 (correcting an oversight in [9]).
In Section 5 we construct the subcaps that we need, deduce that K is a field, construct
the algebra A and show that it is quadratic over K, that it is alternative and that K is in
the center of A. We then show in Section 6 that the cap is projectively equivalent with
V2 (K, A).
But we start with proving in the next section that the examples V2 (K, A) are Veronesean
caps with subdimension dim A.
5
3
Veronesean embeddings are Veronesean caps
In the sequel, K is a field and A is a quadratic alternative division algebra over K.
As already mentioned, V2 (K, A) is an embedding of the projective plane P2 (A) into the
projective space P(V ), where V = K ⊕ K ⊕ K ⊕ A ⊕ A ⊕ A, and where the lines of P2 (A)
correspond to the elliptic spaces. We can view P2 (A) as the projective completion of the
affine plane A2 (A) over A.
The affine plane A2 (A) has A × A as its point set and the following subsets of A × A are
called lines:
[s, t] = {(x, sx + t) | x ∈ A} for s, t ∈ A,
[c] = c × K for c ∈ A.
The special cases [0, 0] and [0] are called the X-axis and Y -axis, respectively. Also, s is
called the slope of the line [s, t], for all s, t ∈ A.
For the projective completion the point at infinity on the lines of slope s ∈ A ∪ {∞} will
be denoted by (s), and the line at infinity is
[∞] := {(s) | s ∈ A ∪ {∞}.
Let (X, Ξ) be the standard Veronesean embedding of P2 (A). Then Theorem 16.3 of
[13] carries over to the general case. In particular, the elliptic spaces are obtained by
intersecting X with a hyperplane of P(V ). If σ is not trivial or char K 6= 2, then we
denote the hyperplane given by {K(x0 , x1 , x2 ; y0 , y1 , y2 ) ∈ K3 × A3 : kx0 + `x1 + mx2 +
ay0σ +y0 aσ +by1σ +y1 bσ +cy2σ +y2 cσ }, k, `, m ∈ K, a, b, c ∈ A, by [k, `, m; a, b, c]. If σ is trivial
and char K = 2, then we denote the hyperplane with point set {K(x0 , x1 , x2 ; y0 , y1 , y2 ) ∈
K3 × A3 : kx0 + `x1 + mx2 = 0}, k, `, m ∈ K, by [k, `, m].
Theorem 3.1 If we identify the points of the projective completion of A2 A with the points
of X in the following way:
(x, y) →
7
K(xxσ , yy σ , 1, y, xσ , xy σ ), x, y ∈ A,
(s) →
7
K(1, ssσ , 0; 0, 0, sσ ), s ∈ A,
(∞) →
7
K(0, 1, 0; 0, 0, 0),
then the elliptic spaces correspond to the lines of P(A) and are determined by intersecting
X with a hyperplane given as follows:
[s, t] →
7
K[ssσ , 1, ttσ ; −t, tσ s, −sσ ],
[c] 7→ K[1, 0, ssσ ; 0, −sσ , 0],
[∞] →
7
K[0, 0, 1; 0, 0, 0],
6
if σ is nontrivial of char K 6= 2, and
[s, t] →
7
K[s2 , 1, t2 ],
[c] →
7
K[1, 0, s2 ],
[∞] →
7
K[0, 0, 1; 0, 0, 0],
if σ is trivial and char K = 2.
The above mapping from the points of P2 (K) to P(V ) will be referred to as the Veronesean
map.
Geometric structure of a line (see 16.2 of [13] for the case K = R.)
Consider the line [∞] of P2 A. The set of points incident with it is
{K(1, ssσ , 0; 0, 0, s) | s ∈ A} ∪ {K(1, 0, 0; 0, 0, 0)}.
This is the quadric of Witt index 1 defined by the equations X0 X1 = Y2 Y2σ and X2 =
Y0 = Y1 = 0, where a generic point has coordinates (X0 , X1 , X2 ; Y0 , Y1 , Y2 ). This is easily
checked to remain true in the inseparable case.
The collineation group of P2 A (see $16.6 of [13] for the case A = O and K = R).
The following permutation leaves the point set of V2 (K, A) invariant.
τ̄ : V → V : K(X0 , X1 , X2 ; Y0 , Y1 , Y2 ) 7→ K(X2 , X0 , X1 ; Y2 , Y0 , Y1 ),
it therefore induces a bijection τ on P2 (A), which is easily seen to be a collineation, called
the triality collineation.
Furthermore, for a, b ∈ A, we define the map τ̃a,b : A3 × K3 → A3 × K3 , which sends
K(X0 , X1 , X2 , Y0 , Y1 , Y2 ) to
K(X0 + Y1σ aσ + aY1 + aaσ X2 , X1 + bY0σ + Y0 bσ + bbσ X2 ,
X2 , Y0 + bX2 , Y1 + aσ X2 , Y2 + Y1σ bσ + aY1σ + abσ X2 )
and which induces the translation Ta,b in A2 (A) given by
τa,b : (x, y) 7→ (x + a, y + b).
The following theorem can be proved exactly as Proposition 17.2 in [13]. Recall that an
elation of a projective plane is a central collineation whose center is incident with its axis.
Theorem 3.2 The group E of elations of P2 A is generated by the translations T and their
2
conjugates T τ and T (τ ) and E acts transitively on the set of non-degenerate triangles of
P2 A.
7
Theorem 3.3 The standard Veronesean embedding (X, Ξ) of P2 A is a Veronesean cap
with k = dimK A.
Proof The lines are (dimK A+1)-dimensional ovoids as just described above. Condition
(V1) is fulfilled by Theorem 3.1. For (V2) we may use the X-axis and the Y -axis, due to
Theorem 3.2. Their point sets are given by the equations
X0 X2 = Y1 Y1σ ,
X1 X2 = Y0 Y0σ ,
and
X1 = Y0 = Y2 = 0,
X0 = Y1 = Y2 = 0,
respectively. Hence their space of intersection is the unique point (0, 0, 1; 0, 0, 0) (which
corresponds to the point (0, 0) of P2 (A)). We use the latter point to verify (V3). First
note that the tangent space at (0, 0, 1; 0, 0, 0) to the X-axis has equations X0 = X1 =
Y0 = Y2 = 0 and the tangent space to the Y -axis at (0, 0) has equations X0 = X1 =
Y1 = Y2 = 0, hence their span T has equations X0 = X1 = Y2 = 0. It is easy to see that
T ∩ X = (0, 0, 1; 0, 0, 0). So, it suffices to show that, for all s ∈ A, all points of the quadric
space ξs ∈ Ξ corresponding to the line [s, 0] are contained in the space Ws spanned by W
and (1, ssσ , 0; 0, 0, sσ ) (the latter corresponds to the point at infinity (s) of the line [s, 0]
of P(A)). But clearly the point (0, 0, 1; sx, xσ , 0) belongs to W , and since
(xxσ , ssσ xxσ , 1; sx, xσ , xxσ sσ ) = (0, 0, 1; sx, xσ , 0) + xxσ (1, ssσ , 0; 0, 0, sσ ),
the assertion follows.
4
Veronesean caps are Moufang projective planes
In this section, we let (X, Ξ) be an arbitrary Veronesean cap in a projective space P(V )
over the skew field K. We always assume that X spans the projective space P(V ). As
already mentioned, the definition of Veronesean cap in [9, 10] is slightly different, and
one of the goal of this section is to show that the present definition is equivalent to the
one in [9, 10] and the fact that P(X, Ξ) is a projective plane. We first prove that every
Veronesean cap has a subdimension.
4.1
The subdimension of a Veronesean cap
If W1 ⊆ W2 are two subspaces of P(V ), then the codimension of W1 in W2 , denoted by
codimW2 W1 , is the dimension of the factor space W2 /W1 , where W1 and W2 are viewed
as subspaces of V . If dim W2 − dim W1 is well-defined, then it is equal to codimW2 W1 .
Lemma 4.1 Every Veronesean cap (X, Ξ) admits a subdimension.
8
Proof Suppose for a contradiction that ξ1 and ξ2 are two elliptic spaces with different
dimensions n1 , n2 , respectively (n1 and/or n2 may be an infinite cardinal). Since by Axiom (V1) the graph with vertex set Ξ, where two vertices are adjacent if the corresponding
elliptic spaces share a point, is connected, we may assume that ξ1 and ξ2 share a point
x ∈ X (by (V2)). Consider an arbitrary elliptic space ξ3 through x, distinct from ξ1
and distinct from ξ2 , and put dim ξ3 = n3 . Without loss of generality we may assume
n1 ≤ n2 ≤ n3 , and n1 < n3 . Axiom (V3) implies that
hTx (ξ1 ), Tx (ξ2 )i = hTx (ξ2 ), Tx (ξ3 )i = hTx (ξ3 ), Tx (ξ1 )i =: U.
Since ξi ∩ ξj = {x}, 1 ≤ i < j ≤ 3, we have codimU Tx (ξ2 ) = n1 − 1 (when considering
U = hTx (ξ1 ), Tx (ξ2 )i) and at the same time codimU Tx (ξ2 ) = n3 − 1 (when considering
U = hTx (ξ3 ), Tx (ξ2 )i). Hence n1 = n3 , a contradiction.
So from now on, we may freely speak about the subdimension of any Veronesean cap.
Also, if x ∈ X, then we denote by Tx the space generated by any pair of tangent spaces
Tx (ξ) and Tx (ξ 0 ) with x ∈ ξ ∩ ξ 0 , ξ, ξ 0 ∈ Ξ and ξ 6= ξ 0 . Next we show the equivalence of
our definition of Veronesean cap with to the one in [9, 10] and the fact that (X, Ξ) with
natural incidence is a projective plane.
4.2
The projective plane associated with a Veronesean cap
Recall that P(X, Ξ) is the geometry having point set X and line set the set Ξ, endowed
with the natural incidence. We will write P = P(X, Ξ) for short. Our goal is to show that
this is a projective plane which satisfies the Moufang condition. We prove some lemmas
to that aim.
We fix an arbitrary elliptic space ξ ∈ Ξ and a point x ∈ X \ ξ. We denote O = X(ξ).
Lemma 4.2 The spaces Tx and ξ are disjoint.
Proof First we note that in the finite case, this is easy. Indeed, if the subdimension is
equal to d (and then d ∈ {1, 2} by Result 48 in [4]), then O has q d + 1 points. Joining
these points to x produces q d + 1 tangent spaces at x of dimension d inside the subspace
Tx , which has dimension 2d, and these meet pairwise in just x. Hence they take into
account (q d + 1)(q d + q d−1 + · · · + q) + 1 points of Tx , which are just all points. The lemma
now follows from (V2).
Now let K be infinite and suppose for a contradiction that some point z belongs to both
Tx and ξ. Pick two distinct points p, q ∈ O (and we can assume that p, q, z are not
collinear), then Tx = hTx ([p, x]), Tx ([q, x])i. Hence there is a line L through z intersecting
Tx ([p, x]) in a point u and intersecting Tx ([q, x]) in a point v. The line hu, xi intersects
Tp ([p, x]) in a point a and the line hv, xi intersects Tq ([q, x]) in a point b. Since |K| > 2 we
find two points a0 ∈ hu, xi and b0 ∈ hv, xi such that z ∈ ha0 , b0 i and a0 6= a, b0 6= b. Since
9
a0 ∈
/ Tp ([p, x]), there is a point a00 ∈ X \ {p} on hp, a0 i and a point b00 ∈ X \ {q} on hq, b0 i.
The line ha00 , b00 i belongs to the 3-space hp, q, z, a0 i, hence it intersects the plane hp, q, zi in
some point z 0 , which consequently belongs to ξ. But ha00 , b00 i ⊆ [a00 , b00 ] and so by (V2), z 0
belongs to X and hence to X([a00 , b00 ]). The latter now contains the three collinear points
a00 , b00 , z 0 , a contradiction.
Now let p ∈ O be arbitrary.
Lemma 4.3 The subspace hO, Tp i does not contain any point of X \ O.
Proof Put W = hO, Tp i. Suppose for a contradiction that u ∈ (W ∩ X) \ O. First note
that Lemma 4.2 immediately yields that O is not contained in Tp ; hence ξ and Tp ([p, u])
intersect in just p and generate W . It follows that ξ and [p, u] also generate W and hence
intersect in a line, contradicting (V2) and the fact that X does not contain lines.
We now choose a subspace Π containing Tx and complementary to ξ (this is possible by
Lemma 4.2). Let ρ be the projection map from ξ onto Π.
Lemma 4.4 The projection ρ is injective on X \ O.
Proof Let, for a contradiction, y, z be two distinct points of X \ O with y ρ = z ρ . Then
ξ is a hyperplane of the subspace hξ, y, zi. Hence the elliptic space [y, z] intersects ξ in
a point of the line hy, zi, which belongs to X by (V2), contradicting (V1) as [y, z] then
would contain three collinear points of X.
We can now prove our first goal.
Proposition 4.5 The geometry P is a projective plane.
Proof Let p be any point of O. By (V3), we have Tp (ξ 0 )ρ = Tp (ξ 00 )ρ , for any ξ 0 , ξ 00 ∈ Ξ
with p ∈ ξ 0 ∩ ξ 00 and ξ 0 6= ξ 6= ξ 00 . We denote by Πp this common projection. It follows
that Πp is a hyperplane in ξ 0 ρ =: Πξ0 , for any ξ 0 ∈ Ξ, p ∈ ξ 0 6= ξ, the complement of which
is the affine space αξ0 := (X(ξ 0 ) \ {p})ρ . Note that, by our choice of Π, the subspace
Πξ0 coincides with Tx (ξ 0 ). We now consider ξ 0 = [x, p]. Choose q ∈ O with q 6= p and
denote ξ 00 = [x, q]. We define αξ00 , Πq and Πξ00 = Tx (ξ 00 ) similarly as for ξ 0 and p. Then
αξ0 ∩ αξ00 = {(ξ 0 ∩ ξ 00 )ρ } = {x}, hence Πp ∩ Πq = ∅. We denote Σ = hΠp , Πq i. It is a
hyperplane in the subspace Tx . Also, Πp and Tx (ξ 00 ) are complementary subspaces in Tx .
Let z be an arbitrary point of Tx \ (Tx (ξ 0 ) ∪ Tx (ξ 00 ) ∪ Σ). Then the subspace hz, Πp i
intersects Tx (ξ 00 ) in a point z 00 . Let u00 ∈ X be the inverse image under ρ of z 00 . Then
the projection of [u00 , p] clearly coincides with hz 00 , Πp i, and so z can be written as uρ with
u ∈ X([u00 , p]). We claim that [x, u] intersects ξ nontrivially.
Indeed, suppose for a contradiction that [x, u] ∩ ξ = ∅. Then, since Tx ([x, u]) ⊆ Tx , since
uρ = z ∈ Tx and since [x, u] = hTx ([x, u]), ui, we have [x, u]ρ ⊆ Tx . Also, Tx ([x, u]) is
10
a hyperplane in [x, u]ρ (since the projection ρ induces an isomorphism from [x, u] into
[x, u]ρ as [x, u] and ξ are disjoint). This implies that [x, u]ρ and Tx (ξ 0 ) intersect in a line L
containing x. This line is not contained in Tx ([x, u]), as Tx ([x, u]) and Tx (ξ 0 ) are disjoint.
Hence L contains a second point y of X([x, u]), y 6= x. By Lemma 4.4, y belongs to Πp .
But that contradicts Lemma 4.3. Hence our claim is proved.
It then follows that z is contained in Tx ([x, u]), and so every point of Tx (ξ 0 ) ∪ Tx (ξ 00 ) ∪ Σ
is contained in a tangent subspace at x to some X-ovoid containing x and intersecting
O in a point. Axiom (V3) implies that there is no room for additional tangent spaces;
hence every elliptic space through x meets ξ nontrivially. Since ξ and x were essentially
arbitrary, this shows that every pair of elliptic spaces intersects nontrivially and hence
(X, Ξ) is a projective plane.
We now go on and prove that P is a Moufang plane.
Proposition 4.6 The projective plane P satisfies the Moufang condition.
Proof We continue with the notation of the previous proof. We know that every point
of Tx \ Πp,q is the projection of a point u of X. Put {r} = [u, x] ∩ ξ. It easily follows
that the projection of Tr ([u, x]) coincides with [u, r]ρ ∩ Πp,q . This now implies that the
projection of X \ ξ onto Π is the André/Bruck-Bose representation [1, 2] of the affine
plane corresponding to P \ O. Hence O is a translation line of P. Since O was arbitrary,
the assertion follows.
A consequence of the André/Bruck-Bose representation in the proof of Proposition 4.6 is
the following.
Corollary 4.7 Every point of P(V ) not contained in an elliptic space is contained in the
span of two elliptic spaces, one of which can be chosen arbitrarily.
Proof Let r be an arbitrary point of P(V ), not contained in any elliptic space. With
the above notation, either rρ is contained in Σ, or coincides with the projection of some
point z ∈ X \ O.
• In the former case, rρ is contained in the projection of Tp , p ∈ O, for a unique point
p ∈ O; hence r is contained in the span of ξ and ξ 0 , where ξ 0 is an arbitrary elliptic
space through p distinct from ξ.
• In the latter case r is contained in the span of ξ and any elliptic space through z.
We now review two special cases.
11
4.3
Veronesean caps with subdimensions 1
If the subdimension of the Veronesean cap (X, Ξ) is equal to 1, then the X-ovoids are
plane ovals. By [10], X is projectively equivalent to a quadric Veronesean variety, i.e.,
a Veronesean cap V2 (K, K) in P5 (K), where K is a commutative field. The geometry
P(X, Ξ) is a projective plane isomorphic to P2 (K). The automorphism groups of the
cap (X, Ξ) (as subgroup of the automorphism group of P(V )) and of the projective plane
P2 (K) are isomorphic in a natural way. We record this and provide a proof.
Proposition 4.8 Every collineation of P2 (K) induces a unique collineation of P5 (K) stabilizing (X, Ξ). Conversely, every collineation of P5 (K) stabilizing (X, Ξ) is induced by
a unique collineation of P2 (K). If moreover |K| > 2, then every collineation of P5 (K)
stabilizing X automatically stabilizes Ξ.
Proof In the case of subdimension 1, the Veronesean map maps the point (x0 , x1 , x2 )
to (x20 , x21 , x22 , x1 x2 , x2 x0 , x0 x1 ). Clearly, these coordinates can be written as the upper
triangular part of the symmetric matrix (x0 x1 x2 )T (x0 x1 x2 ), and we denote the (i, j)entry by xij . Let M = (mij )0≤i,j≤2 be a nonsingular matrix over K, and let θ be an
automorphism of K. Then the collineation sending (x0 x1 x2 ) to (xθ0 xθ1 xθ2 )M induces the
mapping
(x0 x1 x2 )T (x0 x1 x2 ) 7→ M T (xθ0 xθ1 xθ2 )T (xθ0 xθ1 xθ2 )M,
which can be written as
xij 7→
2 X
2
X
mki xθk` m`j .
k=0 `=0
Since this expression is symmetric in i and j, it can be viewed as a semi-linear permutation
of the vector space of all symmetric 3 × 3 matrices over K, which is in a natural way
isomorphic to K6 , and by construction it preserves V2 (K, K) ∼
= (X, Ξ). Also, we claim
that the corresponding collineation of P5 (K) is unique. Indeed, if not, then there would a
nontrivial collineation α of P5 (K) fixing X pointwise. But X clearly contains a skeleton
(a set of 7 points no 6 in the same hyperplane); hence the associated matrix of α is trivial.
Since X contains plane conics also the corresponding field automorphism is trivial, hence
α is trivial after all.
The converse follows from the Fundamental Theorem of Projective Geometry applied to
P2 (K) ∼
= (X, Ξ). Also, if |K| > 2, then the members of Ξ are geometrically characterized as
the only planes containing at least 4 points p1 , p2 , q1 , q2 of X; indeed, if these points would
be contained in a plane not belonging to Ξ, then the intersection x = hp1 , p2 i ∩ hq1 , q2 i
would belong to two distinct members of Ξ, but not to X, contradicting (V2).
The last argument of the previous proof can be recycled to show the following general
result.
12
Corollary 4.9 If |K| > 2, then every Veronesean cap is determined by its point set.
Proof It suffices to show that every quadruple of points p1 , p2 , q1 , q2 ∈ X in a plane is
contained in an elliptic space. This follows as above: if these points would be contained
in a plane not belonging to a member of Ξ, then the intersection x = hp1 , p2 i ∩ hq1 , q2 i
would belong to two distinct members of Ξ, but not to X, contradicting (V2).
4.4
Veronesean caps with subdimension 2
If the subdimension of the Veronesean cap (X, Ξ) is equal to 2, then the X-ovoids are
ovoids in projective 3-spaces. In the classification carried out in [9], in particular in
Subsection 3.2, we determine the equation of a generic X-ovoid, as an elliptic quadric in
P3 (K), K commutative, containing two conics C1 , C2 sharing two distinct points x1 , x2 .
The tangent lines at x1 and x2 to Ci intersect in the point zi , i = 1, 2. In [9], we
coordinatize P3 (K) in such a way that x1 has coordinates (1, 0, 0, 0), x2 is (0, 0, 1, 0), z1
is (0, 1, 0, 0) and z2 is (0, 0, 0, 1). Furthermore, we assume that C1 contains the point
(1, 1, 1, 0), which can be done, but we also assumed in [9] that C2 contains the point
(1, 0, 1, 1), which is not necessarily true since the choice of the point (1, 1, 1, 0) uniquely
determines the first three coordinates of any point (up to a nonzero scalar, though), and
hence we should rather consider the coordinates (1, 0, t, 1). It follows that C1 has equations
2
X1 = X0 X2 ,
X3 = 0,
whereas C2 has equations
tX32 = X0 X2 ,
X1 = 0.
Hence the equation of a generic quadric through C1 and C2 , distinct from the union of the
two planes π1 and π2 , is X12 +kX1 X3 +tX32 = X0 X2 , with k ∈ K. The rest of the argument
in [9] shows that k ∈ K is such that the quadric with equation X12 + kX1 X3 + tX32 =
X0 X2 is elliptic and isomorphic to every X-ovoid. It follows that X12 + kX1 X3 + tX32 is
anisotropic, which means that either this quadratic form defines a unique Galois extension
L of K giving rise to the Hermitian Veronesean cap V2 (K, L) of subdimension 2, or the
characteristic of K is equal to 2, k = 0 and t is a nonsquare
in K. The latter gives rise to
√
0
a purely inseparable quadratic extension K = K( t) of K and we obtain the inseparable
Veronesean cap V2 (K, K0 ) of subdimension 2.
5
The quadratic alternative division algebra A
We now embark on the proof of the Main Result. In this section we show that, with
previous notation, the projective plane P is coordinatized by a quadratic alternative
13
division algebra over K with K in the center. As the cases d = 1, 2 are dealt with in
[9, 10], we may assume d ≥ 3. Consequently, by [4] (Result 48 on page 48), we may
assume that K is infinite.
5.1
General preliminary facts
We take up again the notation of the previous chapter. So ξ ∈ Ξ, O = X(ξ) and ρ is the
projection map from ξ onto a complementary space Π, which we may identify with Tx for
any point x ∈ X \ O. Recall that ρ is injective on X \ O. Also, recall that we have shown
(see Proposition 4.6):
Fact 1 The plane P is a Moufang plane.
The inverse image of Σ under ρ is a hyperplane of P(V ) which we shall denote by T (ξ) or
T (O) and refer to as the tangent space to X at ξ or at O. Clearly, it intersects X in O.
Since Tp ⊂ T (O), Tp ∩ X = {p}. Also, since Πp and Πq are disjoint, we see that the spaces
Tp and ξ 0 , for ξ 0 3 q and ξ 0 6= ξ, generate P(V ), and so are complementary.
Now let x ∈ X \ O (so Tx and ξ are complementary) and let a ∈ X \ {x}. The space
[a, x] meets Tx in Tx ([a, x]) and so the projection of a from Tx onto ξ coincides with the
unique intersection point O ∩ [a, x]. This implies that the image of the projection of
X \ {x} from T (x) onto ξ coincides with O. Denote the projection operator by ρ̃ for
further reference. So the image under ρ̃ of any member of Ξ not containing x coincides
with O. In fact, ρ̃ induces a perspectivity between any pair of lines of P not incident with
the point corresponding to x. By varying x we deduce
Fact 2 All members of Ξ are projectively equivalent. For every ξ ∈ Ξ, the ovoid X(ξ) is
3-transitive.
Note that ρ and ρ̃ are in a certain sense “opposite”. Indeed, we can choose Π = Tx and
then the kernel of one projection is the image of the other. Let ζ ∈ Ξ with x ∈ ζ and
put z = ζ ∩ ξ. Then ζ ∩ Tx = Tx (ζ), and hence, since hz, Tx (ζ)i = ζ and z ∈ ξ, the
image under ρ of ζ, which equals Πζ = hx, Πz i, coincides with Tx (ζ). It follows that S
is projectively equivalent with the set of (d − 1)-spaces arising from the d-spaces Tx (O∗ ),
with O∗ ranging through the set of members of Ξ containing x, by projecting from x onto
Σ (inside Tx ). Consequently,
Fact 3 The tangent spaces at x cover the whole 2d-dimensional tangent space T (x).
14
5.2
Constructing quadric subveroneseans
Now we construct a subset of X which will turn out to be a quadric Veronesean variety.
The construction is completely similar to the one used the classify Hermitian Veronesean
caps, see [9], but we go through the construction again, since we need it for the sequel.
Consider a plane π in Tx through x intersecting Σ in a line not belonging to a spread
element (of S). Let L be a line in π through x. Then L intersects Σ in a point uL , which
belongs to some subspace ΠyL ∈ S, with yL ∈ O uniquely determined. The inverse image
under ρ of L \ {uL } is the intersection of [x, yL ] with hO, Li ⊆ hO, [x, yL ]i. A dimension
argument implies that the inverse image CL0 under ρ of L \ {uL } is the intersection of
X[x, yL ] \ {yL } with the plane hx, uL , yL i. Then CL = CL0 ∪ {yL } is an oval to which the
line uL yL is tangent at yL . Let M be any line in π not containing x and not contained in
Σ. Choose a point z 0 on M \ Σ and let z ∈ X be such that z ρ = z 0 . Then we can apply the
same procedure to M in order to find the inverse image of M \ {v}, where v = M ∩ Σ, and
we also obtain an oval CM . If we do this for three lines not through x and not having a
common intersection point, then we find that the three corresponding ovals span a 5-space
Λ containing all the ovals CL . Let C ∞ be the set of points yL , for L ranging through the
set of lines of π through x. Let V be the union of all CL . Fix an arbitrary oval CL ,
with x ∈ L. We project V inside Λ from the line xyL onto a complementary 3-space W .
Then the ovals through x are projected onto a set R1 of disjoint lines of W and the same
happens for the ovals through yL ; they are projected onto a set R2 of disjoint lines. Now
one sees that R1 is a regulus of a ruled quadric for which R2 is the other regulus except
for one line N . The line N has to be the projection of C ∞ \ {y∞ }, showing that C ∞ is
contained in a plane inside hOi, and is hence also an oval (on O). Consequently we obtain
a system of ovals in Λ with the structure of a projective plane. By Theorem 2.2 of [10],
V is a quadric Veronesean over the commutative field K, and is hence a subplane of P.
Also note that the injectivity of ρ readily implies that hVi ∩ X = V. We record all this
for further reference.
Fact 4 The inverse image on X of every affine plane in Tx whose line at infinity K ∞
is not contained in a member of S, spans a 5-space Λ and is, completed with the oval
C ∞ = Λ ∩ O, isomorphic to a quadric Veronesean. The oval C ∞ coincides precisely with
the set of points z ∈ O with the property Tz ∩ K ∞ 6= ∅. Also, hVi ∩ O = V.
Note also that V is determined by any two of its conics (since these correspond to lines in
an appropriate projection defining a unique plane of PN (K)). We will call every quadric
subveronesean V constructed like this a generic subcap of the first kind.
5.3
Constructing Hermitian Veroneseans containing V
We continue with the notation above. Since P is a Moufang plane, it is coordinatized
by an alternative division ring A, and since we have a subplane isomorphic to P2 (K), the
15
division ring A may be assumed to contain K. Next we want to show that every element
of A is quadratic over K. Therefore we first describe the coordinatization.
We choose O as the line at infinity, and x as the point with coordinates (0, 0). The points
(0), (∞) are then chosen on O and (1, 1) inside V, essentially arbitrary. So K can be
identified with the set of points (r, 0) inside V. An arbitrary element of A corresponds to
a point p = (a, 0), and we may assume a ∈
/ K. The conic of V corresponding to the line
joining (0, 0) and (0) is denoted by C0 ; the one corresponding to the line joining (0, 0)
and (∞) is denoted by C1 . The corresponding lines in π are denoted by M0 and M1 ,
respectively. We also denote y = pρ .
Our next goal is to include V together with p in a Hermitian Veronesean. Such a Veronesean is contained in an 8-space, and we first construct a 7-space in it, which we will call
Ω1 . Then we will extend Ω1 to an 8-space Ω2 .
Let π1 = hM1 , yi, then K1∞ := π1 ∩ Σ is not contained in a member of S since the hx, yi
intersects Σ in a point of the projection of T(0) , and M1 intersects Σ in a point of the
projection of T ((∞)). Hence Fact 4 implies that there is some quadric Veronesean V1 in
X with a conic C1∞ on O such that (V \ C ∞ )ρ = π1 \ K1∞ . Let D be the conic on V1
corresponding to the line hx, yi.
By construction and Fact 4, Ω = hV1 i ∩ hVi contains the 3-space spanned by C1 and the
point (0). If Ω were 4-dimensional, then it would intersect any conic C 0 of V through x
distinct from C1 and from C0 , in a line, which is Tx (C 0 ) by the last assertion of Fact 4.
Hence Tx (C0 ) ⊂ Ω, and consequently C0 ⊂ Ω contradicting V =
6 V1 and the last assertion
of Fact 4. So the space Ω1 generated by V and V1 is 7-dimensional.
We claim that κ := dim(Ω1 ∩ hOi) = 3. Indeed, if κ = 2, then V and V1 would have two
conics in common (C1 and C ∞ ). If κ ≥ 4, then Ω1 ∩ hOi intersects the 6-space hV, Di at
least in a 3-space Π. Then Π intersects the 3-space Π0 spanned by the tangent lines at x
to the conics C0 , C1 and D, which is indeed also contained in hV, Di, in at least one point
q. But then dim((Π0 )ρ ) ≤ 2 contradicting y ∈
/ π.
Next we consider an arbitrary line M2 in π through x but distinct from M0 and M1 . Let
V2 be the quadric Veronesean corresponding to the plane hM2 , yi and note D ⊂ V2 . Then
V2 intersects V1 and V in 3-dimensional spaces Ω21 and Ω2 , respectively, containing x and
(0). If Ω21 ∩ Ω2 were a plane, then Tx (C1 ) ⊂ V2 . As also Tx (C2 ) ⊂ V2 it would follow that
Tx (C0 ) ⊂ V2 . But as (0) ⊂ V2 then C0 ⊂ V2 , a contradiction. Hence Ω21 ∩ Ω2 = hx, (0)i.
Consequently V2 = hΩ21 , Ω2 i (using a dimension argument), and so V2 ⊂ Ω1 . Hence
Fact 5 The inverse image with respect to ρ of the affine 3-space hπ, yi \ Σ is entirely
contained in Ω1 .
Now we extend Ω1 to an 8-space. We use the terminology of a 3-ovoid to mean an ovoid
in a 3-space of some elliptic space, all points of the ovoid are contained in X.
16
Put X1 = X(Ω1 ). Let Q0 = X ∩ hC0 , Di and Q = X ∩ hC ∞ , C1∞ i. Then Q0 and Q are
ovoids of 3-spaces contained in Ω1 . Let O1 be the X-ovoid containing C1 .
We now first prove a general fact, which we state for the original projection ρ, but we will
apply it for different projections.
Fact 6 Let α be a plane through x intersecting a member U of S in a line L. Then the
inverse image of α \ Σ under ρ, completed with the point p∞ of O corresponding to U , is
a 3-ovoid Q, with Q = hQi ∩ X.
Proof Pick 3 non-collinear points pρ1 , pρ2 , pρ3 in α \ L, with pi ∈ X, i = 1, 2, 3. Then the
inverse images of the lines hpρ1 , pρ2 i, hpρ3 , pρ1 i and hpρ2 , pρ3 i determine three conics contained
in the 3-space hp1 , p2 , p3 , p∞ i. It is easy to see that (hp1 , p2 , p3 , p∞ i \ {p∞ })ρ is α \ L.
Putting Q = hp1 , p2 , p3 , p∞ i ∩ X, every point of α \ L is easily seen to be the projection
of a point of Q \ {p∞ }, since Q is an ovoid in a 3-space (as the intersection of a 3-space
with X[x, p∞ ]).
By Fact 6 every affine plane β in hπ, yi with as line at infinity hM0 , yi∩Σ is the projection
of a 3-ovoid Qβ containing (0) and contained in Ω1 by Fact 5. Moreover the union of all
such 3-ovoids is precisely X ∩ Ω1 and each of them intersects C1 (since each plane β
intersects M1 ).
We now project Ω1 \ hO1 i from the elliptic space hO1 i onto T(0) , and obtain a 4-space TΩ
(since T(0) ∩ hO1 i = ∅) and we denote the projection map by ρ1 . Let Σ1 := T (O1 ) ∩ T(0) .
Then the projections of all 3-ovoids Qβ are planes in TΩ pairwise meeting in (0)ρ1 = (0)
and they intersect Σ1 in a set of lines R, which we claim to form a regulus.
Indeed, consider the quadric Veroneseans Vr corresponding to the planes hr, M1 i of Tx ,
with r ∈ K0∞ arbitrary. Since hr, M1 i intersects each plane of hπ, yi through K1∞ in a
line not contained in Σ, we see that Vr contains a conic in every 3-ovoid Qβ (see above).
Since C1 belongs to Vr , the projection of Vr by ρ1 is a plane αr , and by the observation
just made, αr contains a line of each plane Qρβ1 . Hence the line αr ∩ Σ1 intersects every
member of R (and we call such a line an R-transversal ). Since r was arbitrary, the claim
follows.
Consider a point sρ1 in TΩ , with s ∈ X \ X1 . Consider the space Ω2 := hs, Ω1 i. We aim
to show that X2 = X ∩ Ω2 is a Veronesean cap with subdimension 2 (hence of quadratic
Galois extension type of of quadratic inseparable extension type).
Fact 7 Every point t ∈ X such that tρ1 ∈ TΩ belongs to X2 .
Proof Define the points s∞ = h(0), sρ1 i ∩ Σ1 and t∞ = h(0), tρ1 i ∩ Σ1 . There exists
a point u∞ such that both hs∞ , u∞ i and ht∞ , u∞ i contain two points of the regulus R
(consider a plane in Σ1 through the line hs∞ , t∞ i; it intersects R in either a pair of lines,
or a non degenerate conic, then u∞ follows easily). Then the plane αs = h(0), s∞ , u∞ i
17
of TΩ intersects, by construction, two members of R. Hence the corresponding quadric
Veronesean Vαs contains two conics in Ω2 and the additional point s of Ω2 . So Vαs ⊂ Ω2 ;
in particular u ∈ Ω2 . Similarly the latter implies t ∈ Ω2 .
Fact 8 Let a be an arbitrary point in X2 \ C1 and b an arbitrary point in C1 . Then there
is a unique 3-ovoid Qa,b ⊆ X2 containing both a and b. If a = (0), then Qa,b ⊆ X1 .
Proof
Note that aρ1 ∈ TΩ . Also, the projection under ρ1 of Tb contains a unique
element Lb of R. Then the plane through Lb and aρ1 determines the 3-ovoid we were
looking for. The other assertion is obvious.
In particular, there is a 3-ovoid Q2 in X2 containing C2 (taking a ∈ C2 \C1 and b = C1 ∩C2 ).
Let τ be an arbitrary 3-space through Qρ21 not containing (0)ρ1 . Since τ ∩ Σ1 is a plane
β1 which cannot be contained in the union of R, we find a point of (X2 \ X1 )ρ1 by
intersecting τ with the line h(0), c1 i, with c1 ∈
/ β1 , and not on a member of R. We may
assume that this point is sρ1 . Since β1 contains a member, call it K, of R, it contains
a unique R-transversal T1 . Let q = C2 ρ1 ∩ K and select a point b on T1 different from
T1 ∩ K. Select a point a ∈ K different from q and different from K ∩ T1 such that the
plane π 0 := h(0), a, b, sρ1 i ∩ Σ1 does not contain a member of R (this is possible because
there are only two planes through the line hb, sρ1 i for which this does not hold). Now, the
plane πab := ha, b, sρ1 i is contained in τ and intersects Qρ21 in a line A which contains a
and intersects (C2 \ C1 )ρ1 in a point cρ1 different from q (with c ∈ X2 ).
The plane πab determines a quadric Veronesean Vab , which contains exactly two points of
C2 , namely c and x. The line ha, cρ1 i, which is contained in πab , is the projection of a
1
conic belonging to Vab and lies on Q2 . Let e be an arbitrary point on C1 . Then πab ∩ Qρ(0),e
is a point e∗ . Let E ∗ be the set of all such points. If we project E ∗ from (0) onto Σ1 ,
then the projection is obviously contained in the union of R; on the other hand it is also
contained in π 0 ,which contains a and b. Hence the projection of E ∗ is a conic, and since
E ∗ is contained in πab , it is itself a conic containing a and b. Denote the corresponding
set of points of X2 by E2∗ . Note that E2∗ is an infinite set of points intersecting each conic
of Vab in at most 2 points (in fact, it is the intersection of a 4-space in hVab i with Vab , but
we will not need this precise fact).
Now we consider the map ρ2 projecting X \ O2 from hO2 i onto T(0) and let TΩ0 := (Ω1 \
0
hO2 i)ρ2 . The Veronesean Vab projects onto a plane πab
as it has a conic on Q2 . But (E2∗ )ρ1
0
is an infinite set of points intersecting each line of πab
is at most two points; hence it
0
certainly contains a triangle and therefore generates πab
. But since E2∗ ⊆ Ω1 , this implies
0
that πab
is contained in TΩ0 . Consequently sρ2 ∈ TΩ0 . This now implies
Fact 9 The set X2 is independent of the choice of the conic C1 of V through x and
distinct from C0 . Hence we may interchange the roles of C1 and C2 . In particular, if a
is an arbitrary point in X2 \ C2 and b an arbitrary point in C2 , then there is a unique
3-ovoid Q0a,b ⊆ X2 containing both a and b.
18
Now in the 4-space TΩ , the plane Qρ21 intersects every plane containing a member of
R \ {Qρ21 ∩ Σ1 }. Interchanging the roles of C1 and C2 again, this implies that every
3-ovoid Q0a,b as in Fact 9 intersects Q1 , with Q1 the unique 3-ovoid in X2 containing C1 .
Now let c∞ be an arbitrary point of Σ1 not on a member of R. Let M be a line in TΩ
through c∞ containing a point of C2rho1 . Since M contains lots of points off C2ρ1 , Fact 9
and the argument in the first paragraph after Fact 9 imply that M is contained in the
projection πM of a 3-ovoid contained in X2 (and πM is a plane). The intersection πM ∩ Σ1
is a line L∞ containing c∞ and itself contained in a member of S. Hence Fact 6 and
the arbitrariness of c∞ imply that every pair of points of X2 is contained in a unique
3-ovoid contained in X2 , and R together with all lines L∞ define a line spread of Σ1
which determines the André/Bruck-Bose representation of a subplane Ha of P. Hence
X2 , furnished with all 3-ovoids contained in it, is a projective plane. By [9] and Section 4.4,
X2 is a Hermitian or inseparable Veronesean cap. We call such a subcap a generic subcap
of the second kind.
Now Ha contains V and the point (a, 0), and is coordinatized by a field L which contains K.
Since Ha is a subplane of P, we may view L as a subalgebra of A. Since the multiplication
in A can be defined geometrically (see Chapter V of [6]), the element a commutes with
every element of K. Since a was arbitrary, we see that K is contained in the center of
A. Also, we know by [9] and Section 4.4 that L is either a quadratic Galois extension of
K, or L has characteristic 2 and L is a purely inseparable extension of K, with L2 ⊆ K.
In both cases, there is a quadratic equation over K satisfied by a. This implies that A is
quadratic over K. Hence, by Theorem 20.3 of [16], and since we may assume that d ≥ 3,
we deduce that either K is the center of A and A is a quaternion or a Cayley-Dickson
division algebra, or K has characteristic 2 and A is a purely inseparable extension of K
(with A2 ⊆ K). We refer to these two different possibilities as the algebraic and the
inseparable case, respectively.
6
End of the proof
Since A is a quadratic alternative division algebra over K, there is a standard Veronesean
cap (X 0 , Ξ0 ) in some projective space P(V 0 ) associated to it, projectively equivalent to
V2 (K, A). Also, P(X 0 , Ξ0 ) is isomorphic to P. Hence there is a natural bijection : X →
X 0 inducing an isomorphism : P(X, Ξ) → P(X 0 , Ξ0 ). We must extend to a collineation
: P(V ) → P(V 0 ).
Our first task is to show that both and −1 map generic subcaps of each kind to generic
subcaps of the same kind. We denote P 0 = P(X 0 , Ξ0 ).
Lemma 6.1 The bijection and its inverse map generic subcaps to generic subcaps.
Proof With the notation of the previous section, the subcap V is a generic subcap of
the first kind for (X, Ξ). By definition of V2 (K, A), however, the image under of V is
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the quadric Veronesean subcap V2 (K, K), which is a generic subcap of the first kind for
(X 0 , Ξ0 ).
Now we define the relation ∼ on the set of subplanes of P as follows. Two subplanes ∆
and ∆0 are called elementary adjacent, denoted ∆ ∼e ∆0 if they share at least one line and
all of its points. The transitive closure of this relation is called the adjacency relation,
and two subplanes ∆ and ∆0 which are adjacent are denoted ∆ ∼ ∆0 .
Now let ∆ be a generic subcap of the first kind of P, and suppose that ∆0 ∼e ∆. Then ∆
and ∆0 share a point row C, which corresponds to a plane conic (which we also denote by
C) in X. Choose two points x, y in ∆0 not in C. Then the generic subcap of the first kind
determined by C and the unique plane conic through the points x, y and the intersection
of the line xy with C in ∆0 defines a subplane sharing C, x, y, hence they coincide and
so ∆0 is a generic subcap. It follows that every subplane of P adjacent to V is a generic
subcap of the first kind. Now suppose that ∆ is a generic subcap of P. We want to show
that ∆ ∼ V. By considering a generic subcap ∆0 sharing some conic with ∆ and having
two points in V, we obtain ∆0 ∼e ∆ and hence may assume that ∆ and V share at least
two points x, y. Then, considering a generic subcap ∆00 sharing the conic on ∆ through x
and y with ∆, and sharing a conic of V through x not containing y, yields ∆ ∼e ∆0 ∼e V.
So we have shown that the class of subplanes of P adjacent to V coincides with the class
of generic subcaps of the first kind. Similarly, the class of subplanes of P 0 adjacent to V coincides with the class of generic subcaps of the first kind. Since adjacency is defined in
a purely geometric way, it is preserved under and −1 .
The previous lemma says that, if we restrict to an X-ovoid, then it maps plane conics
to plane conics, and similarly for −1 . We will now use this to show that the X-ovoids are
projectively equivalent to the X 0 -ovoids. This is the content of the next lemma, which we
phrase more generally using plane ovals.
Lemma 6.2 Let Π1 and Π2 be two projective spaces of possibly infinite dimension k ≥ 3
which are spanned by ovoids O1 and O2 respectively. Suppose β : O1 → O2 is a bijective
map which maps intersections of O1 with planes of Π1 to intersections of O2 with planes
of Π2 . Then there exists a unique collineation α : Π1 → Π2 extending β.
Proof First, we show uniqueness. Let a ∈ Π1 be an arbitrary point. Then we can
consider three planes which intersect O1 in three different ovals and which have only a
in common. Then the image of a is determined as the unique intersection point of the
images under β of these three ovals.
To prove existence, fix two points x, y ∈ O1 . We will build the mapping in several steps:
Let us fix some notation: for planes π intersecting O1 in an oval O, let π α := hOβ i and
for lines l for which the image under α has already been defined for at least two points
p1 , p2 ∈ l let lα := hpα1 , pα2 i. This definition is unambiguous as we will prove at each step
that α preserves collinearity.
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First we determine the image of all points a ∈ Π1 \ (Tx (O1 ) ∪ Ty (O1 ) ∪ hx, yi) as aα :=
hx, aiα ∩ hy, aiα . We show that collinear points a, b, c ∈ Π1 \ (Tx (O1 ) ∪ Ty (O1 ) ∪ hx, yi)
contained in a line L are mapped onto collinear points.
(a) If dimhL, x, yi = 3 then aα , bα , cα ∈ hx, Liα ∩ hy, Liα .
(b) If β := hL, x, yi is a plane, then consider another plane β 0 through L which intersects
O1 in an oval. Then aα , bα , cα ∈ β α ∩ β 0α .
Moreover, every point of Π2 , which is not contained in hxβ , y β i ∪ Txβ (O2 ) ∪ Tyβ (O2 ),
has a pre-image for α, since it lies on a secant through xβ and a secant through y β .
Next for points p ∈ Tx (O1 ) \ Ty (O1 ), we define pα := Txβ (O2 ) ∩ hy, piα . Clearly, the map is
injective. To check collinearity, first if a, b, c ∈ Tx (O1 ) \ Ty (O1 ) are three collinear points
belonging to a line R, then aα , bα , cα ∈ Txβ ∩ hR, yiα . Next for a, b ∈ Π1 \ (Tx (O1 ) ∪
Ty (O1 ) ∪ hx, yi) and c ∈ Tx (O1 ) \ Ty (O1 ) collinear belonging to a line R. If either x or
y belongs to R, this is immediate. Otherwise, consider the point q = ha, biα ∩ hy, piα .
If q ∈
/ Txβ (L2 ) then by the above paragraph q has a pre-image under α, which must be
ha, bi ∩ hy, pi, a contradiction. Hence q = pα .
For points p in Ty (O1 )\Tx (O1 ) we define pα := Tyβ (O2 )∩hx, piα . The only new case here to
check collinearity is the mixed case, namely when we have three collinear points on a line
R, say a ∈ Tx (O1 )\Ty (O1 ), b ∈ Ty (O1 )\Tx (O1 ) and c ∈ Π1 \(Tx (O1 )∪Ty (O1 )∪hx, yi). But
then since we assume |K| > 2, there exists another point d ∈ Π1 \ (Tx (O1 ) ∪ Ty (O1 ) ∪ hx, yi
on R, and we can proceed as before.
Moreover, now every point in Π2 not contained in hxβ , y β i ∪ (Txβ (O2 ) ∩ Tyβ (O2 )), has a
pre-image for α, with a similar reasoning as before.
For points in Tx (O1 ) ∩ Ty (O1 ) we define pα := hx, piα ∩ hy, piα . Notice that these two lines
do intersect as they are contained in the plane hx, y, piα . Points in Tx (O1 ) ∩ Ty (O1 ) are
mapped to points in m := Txβ (O2 ) ∩ Tyβ (O2 ) and also injectivity is obvious. In order to
check collinearity preservation, suppose that a, b, c are contained in a line R intersecting
Tx (O1 )∩Ty (O1 ) in c. If R contains either x or y, this is immediate, so suppose otherwise. If
R is completely contained in Tx (O1 ), then cα = hx, y, ciα ∩m, which is also the intersection
point of ha, biα with m. If R intersects Tx (O1 ) in p, then ha, biα intersects both Txβ and
Tyβ in points, which do not yet have a pre-image under α. Hence these points belong to m
and must be equal as otherwise R is contained in m. Hence, as in the previous case ha, biα
intersects m and hx, y, piα in a common point, provided R is not contained in hx, y, pi.
But then ha, biα intersect the tangent to xβ in a point, which in the previous steps did
not have a pre-image under α yet. This point must be pα .
Finally, to construct a collineation on the whole of Π1 it remains to define the image of
points z on hx, yi. Let Gz be the set of all lines in Π1 containing z and distinct from hx, yi
and let M1 , M2 ∈ Gz such that M1 , M2 , hx, yi are not conned in a common plane. Then
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there exist two lines N1 , N2 which intersect both M1 and M2 in different points. Hence
also N1α and N2α intersect M1α and M2α in different points. In the plane hN1α , N2α i the lines
M1α and M2α intersect in a point. Since this point does not yet have a pre-image under α
it must be on the line hxβ , y β i. Hence all the images of lines of Gz intersect in this point,
which we define to be z α . Collinearity is obvious for this last case, injectivity is clear by
construction, and surjectivity onto the line hxβ , y β i follows by considering the common
intersection point v of all pre-images of lines containing a point p of such a line, and then
clearly v α = p.
Hence α is a collineation from Π1 to Π2 .
So due to this lemma, we can extend to all points of all elliptic spaces of (X, Ξ). However,
we do not yet know that the resulting map is the restriction of a common collineation
defined globally on P(V ). Before we can tackle that problem, we prove a lemma.
Lemma 6.3 Let O1 , O2 , O3 be three pairwise different lines with pairwise different intersection points xij = Oi ∩ Oj , 1 ≤ i ≤ j ≤ 3 and denote xij = xji . Let z be any point of
X \ (O1 ∪ O2 ∪ O3 ). Put xi = [xjk , x] ∩ Oi , for {i, j, k} = {1, 2, 3} and j ≤ k. Then
hTx12 , x3 i ∩ hTx23 , x1 i ∩ hTx13 , x2 i = {x}.
Proof Let {i, j, k} = {1, 2, 3} and j ≤ k. Since x ∈ [xjk , z] = [xjk , xi ] = hTxij ([xij , xi ]), xi i ⊆
hTxij , xi i, it suffices to show that hTx12 , x3 i ∩ hTx23 , x1 i ∩ hTx13 , x2 i has dimension at most
0.
Let U = hTx12 , x3 i ∩ hTx23 , x1 i ∩ hTx13 , x2 i. We claim that hTx12 , x2 , x3 i ∩ hTx23 , x1 , x2 i =
hz, O2 i. Indeed, both Tx12 , x2 , x3 i and hTx23 , x1 , x3 i contain O2 , hence it suffices to show
that, if we denote by ρ2 the projection operator from hO2 i onto Tz , the projection under
ρ2 of hTx12 , x3 i intersects the projection of hTx23 , x1 i in just z. But this is true since
hTx12 , x3 iρ2 = hTx12 ([x12 , x3 ]), x3 i = [x12 , x3 ] and likewise hTx23 , x1 i = [x23 , x1 ]. The claim
is proved. Consequently U ⊆ hz, O2 i, and similarly, U ⊆ hz, O1 i∩hz, O2 i∩hz, O3 i. Assume
for a contraction that U contains a line L. Then L contains a point ` of O1 , which must
also be contained in hTx23 , x1 i, implying ` = x1 in view of Tx23 ∩ O1 = ∅. Likewise,
x2 , x3 ∈ L. But then the three points x1 , x2 , x3 of X are collinear, a contradiction. Hence
U = {z} and the lemma is proved.
Now we extend to the whole of P(V ). The technique will consist in considering three
X-ovals, extend restricted to these X-ovoids to the elliptic spaces of these X-ovoids,
then extend that map to a collineation defined on the whole of P(V ) and prove it coincides
wit over X.
Theorem 6.4 The bijection : X → X 0 can be uniquely extended to a collineation
P(V ) → P(V 0 ), which maps X onto X 0 and Ξ to Ξ0 .
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Proof
It follows from Proposition 4.8 that we may assume k > 2. First we show
uniqueness. Let α and γ be two collineations which extend . Then by Lemma 6.2, α
and γ coincide on every elliptic space. As every two elliptic spaces intersect in a point it
follows that they also coincide on the span hξ1 , ξ2 i, with ξ1 , ξ2 ∈ Ξ. Corollary 4.7 concludes
the proof.
Next we show existence. Let O1 , O2 , O3 be the X-ovoids with coordinate [0, 0], [0] and [∞],
respectively. Denote hOi i =: ξi ∈ Ξ, i = 1, 2, 3, and denote x12 := O1 ∩ O2 , x13 := O1 ∩ O3
and x23 := O2 ∩O3 (slightly different from the notation in Section 5.3, but more systematic
and suitable for our purposes here).
For every i ∈ {1, 2, 3} there is a collineation i : ξi → hOi i which maps x ∈ Oi to
x . These collineations are induced by semilinear maps. Denote the corresponding Kautomorphisms by θi . We claim that θi = θj , for i, j ∈ {1, 2, 3}. Indeed, the image of V
under is again a generic subcap of the first kind, and i extends the action of to one of
the planes πi of V, i = 1, 2, 3. But i restricted to πi is a collineation, and it is the unique
one extending the action of on the conic X ∩ πi , i = 1, 2, 3. Proposition 4.8 implies
that all i , i = 1, 2, 3, are the restriction of one global collineation hVi → hV i, requiring
θ1 = θ2 = θ3 . The claim is proved.
By recoordinatizing V 0 according to θ1 , or applying θ1−1 to V 0 (which is globally preserved),
we may assume that θ1 is trivial.
Hence, since θ1 = θ3 = id, we can extend 1 and 3 to a map 13 defined on hO1 , O3 i.
By the André/Bruck-Bose representation of P, the domains of definition of 13 and 2
intersects in the line L = hx12 , x23 i. We claim that they coincide on L.
Indeed, let Ci = V ∩ Oi , i = 1, 2, 3. Then 13 /(C1 ∪ C3 ) = /(C1 ∪ C3 ), and 2 /C2 = /C2 .
Denote the unique extension of /V to hVi by 0 . Since extensions of maps on conics to their
planes are unique (as collineations), 13 /L = 0 /L and 2 /L = 0 /L. Hence 13 /L = 2 /L,
proving our claim. Hence there exists a collineation 123 : P(V ) = hξ1 , ξ2 , ξ3 i → P(V 0 )
coinciding with on O1 ∪ O2 ∪ O3 .
Next let z ∈ X \ (O1 ∪ O2 ∪ O3 ). We show that z = z 123 . With xi we denote for all
{i, j, k} = {1, 2, 3} the intersection point [xjk , z] ∩ Li . By Lemma 6.3, z is the unique
intersection point of hTx12 , x3 i, hTx23 , x1 i and hTx13 , x2 i and z is the unique intersection
point of hTx12 , x3 i, hTx23 , x1 i and hTx13 , x2 i. Since 123 is a collineation, we get z = z 123 .
Hence 123 maps X to X 0 , and Corollary 4.9 implies that 123 maps Ξ to Ξ0 .
This completes the proof of the Main Result.
Acknowledgement The second and third author would like to thank the Mathematisches Forschungsinstitut Oberwolfach for providing them with the best possible research
environment imaginable during a stay as Oberwolfach Leibniz Fellow (JS) and Visiting
Senior Researcher (HVM) in June 2013.
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