CUBO, A Mathematical Journal

Vol. 24, no. 02, pp. 307–331, August 2022

DOI: 10.56754/0719-0646.2402.0307

On Severi varieties as intersections of a minimum
number of quadrics

Hendrik Van Maldeghem
1, B

Magali Victoor
1

1Ghent University, Department of

Mathematics: Algebra & Geometry,

Krijgslaan 281, S25, B-9000 Gent,

Belgium.

hendrik.vanmaldeghem@ugent.be B

magali.victoor@ugent.be

ABSTRACT

Let V be a variety related to the second row of the

Freudenthal-Tits Magic square in N-dimensional projective

space over an arbitrary field. We show that there exist

M ≤ N quadrics intersecting precisely in V if and only if

there exists a subspace of projective dimension N −M in the

secant variety disjoint from the Severi variety. We present

some examples of such subspaces of relatively large dimen-

sion. In particular, over the real numbers we show that the

Cartan variety (related to the exceptional group E6(R)) is

the set-theoretic intersection of 15 quadrics.

RESUMEN

Sea V una variedad relacionada a la segunda fila del

cuadrado Mágico de Freudenthal-Tits en el espacio proyec-

tivo N-dimensional sobre un cuerpo arbitrario. Mostramos

que existen M ≤ N cuádricas intersectandose precisamente

en V si y solo si existe un subespacio de dimensión proyec-

tiva N − M en la variedad secante disjunta de la variedad

de Severi. Presentamos algunos ejemplos de tales subespa-

cios de dimensión relativamente grande. En particular, sobre

los números reales, mostramos que la variedad de Cartan

(relacionada al grupo excepcional E6(R)) es la intersección

conjuntista de 15 cuádricas.

Keywords and Phrases: Cartan variety, quadrics, exceptional geometry, Severi variety, quaternion veronesian.

2020 AMS Mathematics Subject Classification: 51E24.

Accepted: 8 June, 2022

Received: 14 October, 2021

c©2022 H. Van Maldeghem et al. This open access article is licensed under a

Creative Commons Attribution-NonCommercial 4.0 International License.

http://cubo.ufro.cl/
http://dx.doi.org/10.56754/0719-0646.2402.0307
mailto:hendrik.vanmaldeghem@ugent.be
https://orcid.org/0000-0002-8022-0040
https://orcid.org/0000-0002-5774-5260
mailto:hendrik.vanmaldeghem@ugent.be
mailto: magali.victoor@ugent.be 


308 H. V. Maldeghem & M. Victoor CUBO
24, 2 (2022)

1 Introduction

It is well known that the Grassmannians of the (split) spherical buildings related to semi-simple

algebraic groups over algebraically closed fields can be described as the intersection of a number

of quadrics, see [7] for the complex case, and [3] and [10] for the more general case. In this paper,

we consider the Grassmannians (or “varieties”) related to the second row of the Freudenthal-Tits

Magic square. Over the complex numbers, these are the so-called “Severi varieties”. However,

these can be considered over any field K (not necessarily algebraically closed anymore), and these

geometries will be also called Severi varieties. A Severi variety lives in a projective space of

dimension N = 5, 8, 14 or 26 and is the set-theoretic and scheme-theoretic intersection of N + 1

quadrics, the equations of which carry a particularly elegant combinatorics, see [11]. The question

we’d like to put forward in this paper is whether we can describe the Severi varieties set-theoretically

with fewer quadrics, and ultimately try to find the minimum number of quadrics the intersection

of which is precisely the given Severi variety. Our motivation is entirely curiosity and beauty; the

latter under the form of a rather unexpected connection we found.

We will show that the N + 1 quadrics referred to above are linearly independent from each other.

Also, every quadric containing the given Severi variety is a linear combination of these N + 1

quadrics. These two facts point, in our opinion, to the conjecture that no set of N quadrics can

intersect precisely in the Severi variety. However, the quadric Veronese surface (the case N = 5

Severi variety) over fields of characteristic 2 is the set-theoretic intersection of three quadrics, see

Lemma 4.20 in [6]. Moreover, it was stated in [2], however without proof, that in the case N = 8,

the Severi variety is the set-theoretic intersection of only 6 quadrics. Hence the above conjecture

is false. In general, we will show the following equivalence:

Main Result. There exist M ≤ N quadrics intersecting precisely in the given Severi variety ⇐⇒

there exists a subspace of projective dimension N −M in the secant variety disjoint from the Severi

variety.

A more detailed and precise statement will be provided in Section 3. In fact, that statement and

its proof allow one, in principle, to describe all equivalence classes of systems of M ≤ N quadratic

equations exactly describing a given Severi variety. As an application, we will do this explicitly

in the simplest case, N = 5. For the other cases we content ourselves with giving examples for

relatively small M. In particular we will exhibit the real Cartan variety (the Grassmanian of

type E6,1 in 26-dimensional real projective space) as the intersection of only 15 quadrics (whereas

initially, we had 27 of them). It would require additional methods and ideas to pin down the

minimal M for each case and each field, so we consider that to be out of the scope of this paper.

About the method of our proof: Usually, the equations of the N + 1 initial quadrics are partial

derivatives of a cubic form (which has to be taken for granted). In the present paper, we start



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On Severi varieties as intersections of a minimum number of quadrics 309

with the combinatorics of the equations of the quadrics and derive the cubic form from that. This

enables us to make a few geometric observations and interpretations which lead to a proof of the

Main Result.

Since the secant variety of a Severi variety always contains at least one point outside the variety,

we recover in our special case of Severi variety already the general result of Kronecker saying that

any projective variety in PN
K

is a set theoretic intersection of (at most) N hypersurfaces (in our

case quadrics), see Corollary 2 in [5]. One could also ask the equivalent question for the scheme-

theoretic intersection of quadrics, but we did not consider that. It seems to us that the answer we

give in the present paper for the Segre variety is also valid in the scheme-theoretic sense, but the

minimal examples for the line Grassmannian and the Cartan variety are not.

2 Preliminaries

2.1 The varieties

The main objects in this paper are the quadric Veronese surface V2(K) over any field K, the

Segre variety S2,2(K) corresponding to the product of two projective planes over K, the line

Grassmannian G2,6(K) of projective 5-space over K, and the Cartan variety E6(K) associated to

the 27-dimensional module of the (split) exceptional group of Lie type E6 over the field K. These

varieties can be defined as intersections of quadrics (and we will do so in Subsection 4.1 below),

but it might be insightful to also have the classical definition, which we now present. In what

follows, K is an arbitrary field and PN
K

or PN denotes the N-dimensional projective space over K,

which we suppose to be coordinatized with homogeneous coordinates from K after an arbitrary

choice of a basis.

The quadric Veronese surface V2(K)—This is the image of the Veronese map ν : P
2 → P5 :

(x, y, z) 7→ (x2, y2, z2, yz, zx, xy).

The Segre variety S2,2(K)—This is the image of the Segre map P
2×P2 → P8 : (x, y, z; u, v, w) 7→

(xu, yu, zu, xv, yv, zv, xw, yw, zw).

We may view the set of 3 × 3 matrices over K as a 9-dimensional vector space, and the set of

symmetric 3 × 3 matrices as a 6-dimensional subspace. Then we may consider the corresponding

projective spaces of (projective) dimension 8 and 5, respectively, in the classical way by considering

the 1-spaces as the points. In this way, the Segre variety S2,2(K) corresponds exactly with the

rank 1 matrices; explicitly

K(xu, yu, zu, xv, yv, zv, xw, yw, zw) ↔ K









xu yu zu

xv yv zv

xw yw zw









.



310 H. V. Maldeghem & M. Victoor CUBO
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Similarly, the quadric Veronese surface V2(K) corresponds exactly with the rank 1 symmetric

matrices; explicitly

K(x2, y2, z2, yz, zx, xy) ↔ K









x2 yx zx

xy y2 zy

xz yz z2









.

In particular, V2(K) is a subvariety of S2,2(K) obtained by intersecting with a 5-dimensional

subspace.

There exist other Segre varieties; in general Sn,m(K) is defined as the image in P
nm−1 of the map

(xi, yj)1≤i≤n,1≤j≤m 7→ (xiyj)1≤i≤n,1≤j≤m. The images of the marginal maps defined by either

fixing the xi, 1 ≤ i ≤ n, or the yj, 1 ≤ j ≤ m, are called the generators of the variety (in case of

S2,2(K) the generators are 2-dimensional projective subspaces).

The line Grassmannian G2,6(K)—Denote the set of lines of P
5, or equivalently, the set of 2-

spaces of K6 by
(

K
6

K2

)

. Then G2,6(K) is the image of the Plücker map

(

K
6

K2

)

→ P14 : 〈(x1, x2, . . . , x6).(y1, y2, . . . , y6)〉 7→ (xiyj − xjyi)1≤i<j≤6.

Denote the coordinate of P14 corresponding to the entry xiyj − xjyi by pij, 1 ≤ i < j ≤ 6. By

restricting to y1 = y2 = y3 = x4 = x5 = x6 = 0, we see that S2,2(K) is a subvariety of G2,6(K)

obtained by intersecting with an 8-dimensional projective subspace with equation p12 = p13 =

p23 = p45 = p46 = p56 = 0.

The Cartan variety E6(K)—This variety is traditionally defined using a trilinear or cubic form,

and we postpone this to Subsection 4.1. It is an exceptional variety in the sense that it cannot be

defined, using classical notions like Plücker or Grassmann coordinates, from a projective space.

The above varieties share the following properties, see [9]. Set N = 2+3M, with M = 1, 2, 4, 8. Let

V be one of the varieties V2(K), S2,2(K), G2,6(K) or E6(K), in P
N, with M = 1, 2, 4, 8, respectively.

Then there exists a unique set H of (M + 1)-dimensional subspaces, called host spaces, satisfying

(1) every pair of points of V is contained in at least one host space;

(2) the intersection of V with any host space is a non-degenerate quadric of maximal Witt index

in the host space.

Borrowing some terminology from the theory of parapolar spaces, we shall refer to the quadrics in

(2) as symps. Also, we shall call two points of the variety collinear when all points of the joining

projective line belong to the variety.

If we specialize K = C, then V is sometimes called a Severi variety; these are the only complex

varieties with the property that their secant varieties are not the whole projective space, but

the secant variety of every variety of the same dimension in a lower dimensional projective space



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On Severi varieties as intersections of a minimum number of quadrics 311

coincides with the ambient space. So we will also refer to these varieties over an arbitrary field as

the Severi varieties.

2.2 A generalized quadrangle

The introduction of an appropriate cubic form and explicit descriptions using coordinates will be

greatly facilitated by using the language of finite generalized quadrangles. A finite generalized

quadrangle (of order (s, t)) is an incidence system Γ = (P, L ) of finitely many points (P) and

lines (L ), where each line is a subset of P, such that each line contains s + 1 points, through

each point pass t + 1 lines, and for each point p and each line L with p /∈ L, there exists a unique

point-line pair (q, M) such that p ∈ M and q ∈ L ∩ M. We are only interested in generalized

quadrangles of order (2, t), and then, by 1.2.2 and 1.2.3 of [8], necessarily t ∈ {1, 2, 4}. Moreover,

by 5.2.3 and 5.3.2 of [8], for each t ∈ {1, 2, 4}, there is a unique generalized quadrangle GQ(2, t) of

order (2, t) and GQ(2, 1) is contained in GQ(2, 2) as a subgeometry, and GQ(2, 2) is contained in

GQ(2, 4) as a subgeometry.

In the rest of this paper, we denote by Γ = (P, L ) the generalized quadrangle GQ(2, 4). An

explicit construction of Γ runs as follows, see Section 6.1 of [8]. Let P′ be the set of all 2-subsets

of the 6-set {1, 2, 3, 4, 5, 6, }, and define

P = P′ ∪ {1, 2, 3, 4, 5, 6} ∪ {1′, 2′, 3′, 4′, 5′, 6′}.

Denote briefly the 2-subset {i, j} by ij, for all appropriate i, j. Let L ′ be the set of partitions of

{1, 2, 3, 4, 5, 6} into 2-subsets and define

L = L ′ ∪ {{i, j′, ij} | i, j ∈ {1, 2, 3, 4, 5, 6}, i 6= j} .

Then Γ = (P, L ) is a model of GQ(2, 4). The subgeometry Γ′ = (P′, L ′) is a model of GQ(2, 2).

Further restriction to

P
′′ = {ij | i ∈ {1, 2, 3}, j ∈ {4, 5, 6}},

with induced line set

L
′′ = {{14, 25, 36}, {15, 26, 34}, {16, 24, 35}, {14, 26, 35}, {15, 24, 36}, {16, 25, 34}},

produces a model Γ′′ = (P′′, L ′′) of GQ(2, 1), which we sometimes refer to as a 3 × 3 grid.

The sets {1, 2, 3, 4, 5, 6} and {1′, 2′, 3′, 4′, 5′, 6′} have the property that they both do not contain

any pair of collinear points, and that non-collinearity is a paring between the two sets. Such a pair

of 6-sets is usually called a double six.

Finally, we need the notion of a partial spread, which is just a set of disjoint lines. A spread is a

partial spread that partitions the point set. Every generalized quadrangle of order (2, t), t = 1, 2, 4,

satisfies the following property (again, see Section 6.1 of [8]):



312 H. V. Maldeghem & M. Victoor CUBO
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(∗) Every pair of disjoint lines is contained in a unique generalized subquadrangle of order (2, 1)

Three mutually disjoint lines of a subquadrangle of order (2, 1) will be called a regulus. Property

(∗) can be reformulated as “every pair of disjoint lines is contained in a unique regulus”. A partial

spread which is closed under taking reguli of pairs of its members is called regular. The GQ(2, 4)

contains regular spreads; a maximum regular partial spread of GQ(2, 2) has size 3, and obviously

the GQ(2, 1) contains exactly two regular spreads. In this paper, we will fix the following regular

spread S of Γ, which induces maximum regular partial spreads in Γ′ and Γ′′:

S = {{14, 25, 36}, {15, 26, 34}, {16, 24, 35}, {12, 2, 1′}, {23, 3, 2′}, {13, 1, 3′},

{45, 4, 5′}, {56, 5, 6′}, {46, 6, 4′}} .

The lines {14, 25, 36}, {15, 26, 34}, {16, 24, 35} form a maximum regular partial spread in both Γ′

and Γ′′.

3 Main result

In this paper, we prove the following connection between the minimum number of quadrics needed

to describe a Severi variety and the largest dimension of a projective subspace in the secant variety

disjoint from the variety itself.

Theorem 3.1. Let V be either the quadratic Veronese surface V2(K), the Segre variety S2,2(K),

the line Grassmannian G2,6(K), or the Cartan variety E6(K), in N-dimensional projective space

P
N over K, with N = 5, 8, 14, 26, respectively. Then V is the intersection of N − d quadrics and

no less, where d is the dimension of a maximum dimensional projective subspace of PN entirely

consisting of points lying on a secant of V , or in the nucleus plane if V = V2(K) with char K = 2,

but not on V . More precisely, the equivalence classes of the systems of N − d linearly independent

quadrics intersecting precisely in V are in natural bijective correspondence with the d-dimensional

projective subspaces of PN entirely consisting of points lying on a secant of V , or in the nucleus

plane if V = V2(K) with char K = 2, but not on V .

To fix the ideas, we provide a full proof for the variety E6(K). The other cases are completely

similar. We comment on them along the way, if differences arise.

4 Proof of Theorem 3.1

4.1 A cubic form

The Cartan variety E6(K) is the intersection of 27 well chosen degenerate quadrics. The equations

of these quadrics can be described as follows. Let K27 be the vector space underlying P26 and denote



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On Severi varieties as intersections of a minimum number of quadrics 313

by 〈v〉 the point of P26 corresponding to the nonzero vector v ∈ K27. Recall that Γ = (P, L )

is the generalized quadrangle of order (2, 4) and S is a regular spread of Γ. Label the standard

basis vectors of K27 with the points of Γ; so the standard basis is {ep : p ∈ P}. Each point p ∈ P

defines a unique quadratic form Qp given in coordinates by

Qp(v) = Xq1Xq2 −

∑

{p,r1,r2}∈L \S

Xr1Xr2,

where {p, q1, q2} ∈ S . Now define the map φ : K
27 → K27 : v 7→ (Qp(v))p∈P. Our basic

observation is the following identity.

Observation 4.1. For all v ∈ K27 we have φ(φ(v)) = C(v)v, where

C(v) =

∑

{p,q,r}∈S

XpXqXr −

∑

{p,q,r}∈L \S

XpXqXr.

Also, φ(v) = ∇C(v) (the gradient in the classical sense).

Proof. The last assertion is obvious. We show the first one. We have to prove the following identity

for each point p ∈ P:

Qq1(v)Qq2(v) −

∑

{p,r1,r2}∈L \S

Qr1(v)Qr2(v) = C(v)Xp, (4.1)

where {p, q1, q2} ∈ S and v = (Xq)q∈P. Since each Qq(v), q ∈ P, has five terms of degree 2 in

the coordinates of v, the above sum has 125 terms of degree 4. Since each Qq(v) has a unique term

containing Xp, there are five terms containing X
2
p and another 40 containing Xp but not X

2
p. The

terms with X2p are easily seen to be

X2pXq1Xq2 −

∑

{p,r1,r2}∈L \S

X2pXr1Xr2. (4.2)

For each line {q1, s, s
′} ∈ L , we have the combined terms XpXq1 of Qq2(v) and −XsXs′ of Qq1(v),

resulting in a term −XpXq1XsXs′ in the left hand side of Equation (4.1). Note that {q1, s, s
′} /∈ S .

Similarly for the lines through q2. We conclude that the terms of Qq1Qq2 containing Xp but not

X2p are given by

−

2
∑

i=1

∑

{qi,s,s
′}∈L \S

XpXqiXsXs′. (4.3)

Now let r ∈ P be collinear to p but distinct from q1 and q2. Let {r, s, s
′} ∈ L , with p /∈ {s, s′}.

First suppose that {r, s, s′} ∈ S . Let r′ ∈ P be such that {p, r, r′} ∈ L \ S . Then have the

combined terms −XpXr of Qr′(v) and XsXs′ of Qq1(v), resulting in a term −XpXrXsXs′ in the

left hand side of Equation (4.1). If {r, s, s′} ∈ L \ S , then we obtain the same term, but with



314 H. V. Maldeghem & M. Victoor CUBO
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the opposite sign. These terms, together with those of Expressions (4.2) and (4.3) already provide

the full right hand side of Equality (4.1). The remaining 125 − 5 − 40 = 80 terms in the left

hand side of Equality (4.1) should now cancel pairwise. Disregarding the signs, they are all of the

form Xs1Xs′1Xs2Xs
′

2
, where {ri, si, s

′
i} ∈ L , i = 1, 2, with {p, r1, r2} ∈ L . These three lines are

contained in a unique grid









p r1 r2

q s1 s2

q′ s′1 s
′
2









, for some q, q′ ∈ P,

where the rows and columns correspond to lines of Γ. Hence in the term Qq(v)Qq′ (v) also appears a

term Xs1Xs2Xs′1Xs
′

2
, up to sign. We now have to see that the signs are opposite. If {p, r1, r2} ∈ S ,

then both signs are +, but the terms nevertheless cancel since Qq(v)Qq′ (v) appears with a minus

sign in Equality (4.1). Note that it does not make any difference whether {q, s1, s2} ∈ S or not,

since, by the regularity property of S we have {q, s1, s2} ∈ S if and only if {q
′, s′1, s

′
2} ∈ S .

Now suppose {p, r1, r2} ∈ L \ S . We may also assume that {p, q, q
′} ∈ L \ S , as otherwise we

are back in the previous case by interchanging the roles of {r1, r2} and {q, q
′}. If exactly one of

the other lines of the grid belongs to the spread S , then the signs are opposite. The regularity

of S implies that at most one other line belongs to S ; we now claim that every 3 × 3 grid of Γ

contains at least one spread line. Indeed, we count 12 grids with three spread lines and 9·12 = 108

grids with a unique spread line. In total there are 45 lines, each in 16 grids, but each also counted

6 times. Hence there are 120 3 × 3 grids in total, which shows our claim and the observation. ✷

Comments on the other cases.

(i) The Grassmannian variety G2,6(K) arises from the Cartan variety above by setting Xp = 0

for all points p in a double six. Indeed, the analogue of the construction above considers Γ′

in place of Γ and a maximal regular partial spread S ′ in place of S (S ′ consists just of

three disjoint lines of a grid). That this works can be seen through the model of Γ, Γ′ and

S given in Subsection 2.2. Since G2,6(K) is the intersection of all quadrics with equation

pijpkℓ + pikpℓj + piℓpjk = 0 (as follows from Theorem 3.8 in [6]), it suffices to make a choice

between each pij and pji in order to get the signs lined up with the above rule and the choice

of S ′. But this can simply be done by retaining pij for i ∈ {1, 2, 3} and j ∈ {4, 5, 6}, and

(ij) ∈ {(12), (23), (31), (45), (56), (64)}, as an elementary calculation shows.

(ii) The Segre variety S2,2(K) arises from the Cartan variety by setting Xp = 0 for all points

outside a regulus of spread lines. This can easily be seen through the construction in Sub-

section 2.1, denoting the point of the grid associated to the entry (i, j) in the 3 × 3 ma-

trix by qij and the corresponding coordinate by xij, we let the grid be defined by the

lines {qij, qkℓ, qmn} with {i, k, m} = {j, ℓ, n} = {1, 2, 3}. If we choose the spread lines as



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On Severi varieties as intersections of a minimum number of quadrics 315

{q11, q22, q33}, {q12, q23, q31} and {q13, q32, q21}, then we see that Qp is exactly the co-factor

of the entry corresponding to p in the matrix (xij)1≤i,j≤3. This indeed defines S2,2(K) as

can be deduced from Theorem 4.94 in [6], or from [2].

(iii) The quadric Veronese variety V2(K) arises from the Cartan variety by setting Xp = 0 for

all points outside a regulus {L1, L2, L3} of spread lines and Xp1 = Xp2 for collinear points

pi ∈ Li, i = 1, 2. Indeed, in the previous paragraph, choosing L3 = {q11, q22, q33}, collinear

points outside this line correspond to symmetric entries of the matrix. Here, the gradient

is not identical to φ; the last three coordinates of the gradient are twice the last three

coordinates of φ, hence there is special behaviour in characteristic 2.

Denoting by v.w the ordinary dot product of v and w in K27, we observe the following.

Observation 4.2. For arbitrary v, w ∈ K27 and t ∈ K, we have

C(v + tw) = C(v) + tφ(v).w + t2v.φ(w) + t3C(w). (4.4)

Proof. It is clear that the coefficient of t0 and t3 are C(v) and C(w), respectively. It remains to

explain the coefficient of t, as the one of t2 is obtained by switching the roles of v and w. Now,

obviously, the coefficient of t is linear in w, hence if suffices to set w = ep for p ∈ P. Then we see

that the coefficient of t in C(v + tep) is equal to
∂C(v)
∂Xp

ep = Qp(v)ep. Now Identity (4.4) follows. ✷

Hence we deduce that the adjoint square v♯ in the sense of Aschbacher [1], is, up to reordering

the coordinates, exactly equal to φ(v). Hence C(v) is the cubic form related to E6(K) and the

Chevalley group E6(K) acts on P
26 with three orbits, which are easily seen to be defined as

(i) the points of the variety E6(K), namely those corresponding to the vectors v with φ(v) = ~o.

These points are the white points;

(ii) the points off the variety E6(K) corresponding to the vectors v with C(v) = 0. These points

are the grey points;

(iii) the points corresponding to vectors v with C(v) 6= 0. These points are the black points.

We have taken the notions of white, grey and black from Aschbacher [1]. See also Cohen [4] for a

very comprehensive introduction.

Comments on the other cases. For the quadric Veronese variety V2(K) the group has more

than three orbits; in this case, and if char K = 2, the grey points also comprise all points of the

nucleus plane.

It now follows from (i), (ii) and Identity (4.4) that the projective null set of the cubic form C is

exactly the secant variety of E6(K).



316 H. V. Maldeghem & M. Victoor CUBO
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Observation 4.3. Let v be a nonzero vector of K27.

(i) The point 〈v〉 is a white point if and only if φ(v) = ~o;

(ii) the point 〈v〉 is grey if and only if φ(v) 6= ~o and the point 〈φ(v)〉 is white;

(iii) the point 〈v〉 is black if and only if 〈φ(v)〉 is a black point.

Proof. (i) This follows immediately from the definition of white points above.

(ii) By definition, the point 〈v〉 is grey if and only if φ(v) 6= ~o and C(v) = 0. The latter is

equivalent to φ(φ(v)) = ~o, which is equivalent to φ(v) being white by (i).

(iii) Suppose 〈v〉 is black. If 〈φ(v)〉 is white or grey, then C(φ(v)) = 0, implying φ(φ(φ(v))) = ~o.

But the left hand side is equal to φ(C(v)v) = C(v)2φ(v) 6= ~o, a contradiction. Now suppose

φ(v) is black. Then φ(φ(φ(v))) is a non-zero multiple of φ(v), and so φ(φ(v)) cannot be equal

to ~o, implying C(v) 6= 0 and 〈v〉 is black. ✷

It follows from the previous observation that 〈φ(v)〉 is never a grey point. We record this for further

reference.

Corollary 4.4. For each v ∈ K27, 〈φ(v)〉 is never a grey point.

We also observe that transitivity of the automorphism group of E6(K) implies the following.

Observation 4.5. Let v be a nonzero vector of K27. Then 〈v〉 is a white point if and only if there

exists a grey point 〈w〉 with 〈φ(w)〉 = 〈v〉.

Proof. If 〈w〉 is grey, then by Observation 4.3 (ii), 〈φ(w)〉 is white. Now let 〈v〉 be a white point.

Let 〈w0〉 be a grey point (for instance the point 〈ep + eq〉 with p and q collinear points of Γ). Then

by Observation 4.3 (ii), 〈φ(w0)〉 is white. Let g be an automorphism of E6(K) mapping 〈φ(w0)〉 to

〈v〉. Then 〈w
g
0〉 is grey and 〈φ(w

g
0)〉 = 〈φ(w0)

g = 〈v〉. ✷

Observation 4.6. For every white point 〈v〉, the set {〈w〉 | ~o 6= φ(w) ∈ 〈v〉} is the set of grey

points of a (9-dimensional) host space of P26 (hence generated by the points of some fixed symp).

Proof. Let p ∈ P be arbitrary. Let 〈w〉 be a grey point belonging to the host space Up := 〈eq |

p ⊥ q ∈ P〉. Then clearly φ(w) is a nonzero multiple of ep. By transitivity of the automorphism

group, we thus see that for every white point 〈v〉, the set {〈w〉 | ~o 6= φ(w) ∈ 〈v〉} is the set of grey

points of a union of host spaces of P26. Suppose that we have the union of at least two host spaces.

By transitivity, we may assume that two of these host spaces are Up and Uq, with p, q ∈ P. But

we already know that these map to 〈ep〉 and 〈eq〉, respectively, which are distinct. The assertion

now follows. ✷



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On Severi varieties as intersections of a minimum number of quadrics 317

4.2 A lemma

Lemma 4.7. Let Q be a quadratic form whose null set contains E6(K). Then Q is a linear

combination (with constant coefficients in K) of the Qp, p ∈ P.

Proof. Let Q be given by the polynomial

Q(v) =
∑

{p,q}⊆P

a{p,q}XpXq,

with a{p,q} ∈ K. Since all points corresponding to the standard basis vectors belong to E6(K), we

have a{p} = 0, for all p ∈ P. Now let p, q ∈ P be distinct but non-collinear in Γ. Then one easily

checks that 〈ep + eq〉 ∈ E6(K). Hence the coefficient a{p,q} of XpXq in Q(v) is also 0.

Now consider a line L ∈ S and a line M ∈ L \ S with L ∩ M = {p}, p ∈ P. Let L = {p, q1, q2}

and M = {p, r1, r2}. Then clearly the point 〈eq1 + eq2 + er1 + er2〉 belongs to E6(K). This implies

that a{q1,q2} = −a{r1,r2} =: ap. Now it is clear that Q(v) =
∑

p∈P

apQp(v), proving the lemma. ✷

Noting that, for collinear points q1, q2 ∈ P, the vector eq1 + eq2 belongs to the null set of each

quadratic form Qp, p ∈ P, except for the unique point p with {p, q1, q2} ∈ L , we see that

Observation 4.8. The set {Qp : p ∈ P} is a linearly independent set of quadratic forms and no

proper subset of it intersects precisely in E6(K).

Comments on the other cases. Care has to be taken for the case V2(K), not only since the

automorphism group can have more than three orbits on the points (and on the hyperplanes) of

the surrounding projective space, but also since this case behaves in an exceptional way for small

fields. Let us provide some quick details. With respect to the representation given as definition in

Subsection 2.1, we have

φ(x1, x2, x3, x23, x31, x12) =

(x2x3 − x
2
23, x3x1 − x

2
31, x1x2 − x

2
12, x31x12 − x1x23, x12x23 − x2x31, x23x31 − x3x12),

and

C(x1, x2, x3, x23, x31, x12) = x1x2x3 + 2x12x23x31 − x1x
2
23 − x2x

2
31 − x3x

2
12.

Observations 4.5 and 4.6 need an alternative proof, since the group does not act transitively on

the grey points. However, one calculates easily that

φ(0, 0, 0, k, −ℓ, 0) = (−k2, −ℓ2, 0, 0, 0, −kℓ) = −ν(k, ℓ, 0)

and

φ(1, 1, a2 + b2, −b, −a, 0) = (a2, b2, 1, b, a, ab) = ν(a, b, 1),



318 H. V. Maldeghem & M. Victoor CUBO
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which covers all points of V2(K). This shows the nontrivial direction of Observation 4.5. Obser-

vation 4.6 follows from a similar calculation and is left to the reader. Lemma 4.7 only holds for

fields with at least four elements. To prove this, one just expresses that a generic point ν(x, y, z)

satisfies a quadratic equation, and one argues that, if the field has at least 4 elements, then the

corresponding quadratic form is a linear combination of the XiXj − X
2
ij, ij ∈ {12, 23, 31}, and the

XiXjk −XkiXij, ijk ∈ {123, 231, 312}. For |K| ≤ 3, all points of V2(K) satisfy X1X23 −X2X23 = 0

since x = x3 for all x ∈ K, and this is not a linear combination of the basic quadratic equations.

Finally, Observation 4.8 is false, see Subsection 5.2 below.

4.3 Reducing the number of quadrics—End of the proof

Lemma 4.7 and Observation 4.8 indicate that we need all 27 quadratic forms to describe E6(K) as

the intersection of quadrics. However, making suitable linear combinations, we can actually reduce

the number of quadrics. To do this, let U be a subspace of K27 such that all its non-zero vectors

correspond to grey points, and we use the same notation U for the corresponding subspace of P26.

Let {Hi : i ∈ I} be a minimal set of hyperplanes of K
27 whose intersection is exactly U (then

|I| + dim U = 27, where dim U is the vector dimension of U). For each

Hi ↔
∑

p∈P

a(i)p Xp = 0, i ∈ I,

define the quadratic form Qi given by

Qi(v) =
∑

p∈P

a(i)p Qp(v).

Note that for a vector v ∈ K27 we have Qi(v) = 0 if and only if φ(v) ∈ Hi.

Clearly, the null set of each Qi contains the vectors corresponding to E6(K). Conversely, suppose

some nonzero vector v belongs to the null set of each Qi, i ∈ I. Then, by construction, 〈φ(v)〉 ∈ U.

If φ(v) 6= ~o, this would mean that 〈φ(v)〉 is a grey point, contradicting Corollary 4.4.

Conversely, suppose E6(K) is the intersection of the null sets of a number of quadratic forms Qi,

i ∈ I. By Lemma 4.7, each quadratic form Qi is a linear combination of the Qp, p ∈ P, say

Qi(v) =
∑

p∈P

a(i)p Qp(v), a
(i)
p ∈ K.

For i ∈ I, let the hyperplane Hi be given by the equation

Hi ↔
∑

p∈P

a(i)p Xp = 0.

Suppose there is a white or black point 〈v〉 contained in each hyperplane Hi, i ∈ I. Then Lemma 4.5

and the definition of C(v) implies that there exists w ∈ K27 with φ(w) = v and with 〈w〉 grey or



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On Severi varieties as intersections of a minimum number of quadrics 319

black. It follows that w is in the null set of each Qi, i ∈ I, contradicting the fact that 〈w〉 is not

white. This actually shows the last claim of Theorem 3.1, and the first claim also follows.

Hence the minimum number of quadrics completely describing E6(K) as their intersection is equal

to 27 − d′, where d′ = d + 1 is the dimension of a maximum dimensional subspace containing no

vectors corresponding to white or black points.

5 Examples and applications

In this section, we determine the exact value of d for some specific cases. Our results will show

that d strongly depends on the field K and therefore the determination of d for every field K is

beyond the scope of this paper.

We begin with some general observations.

5.1 General observations

To ease notation, we will identify the projective version of φ with φ, i.e., we will write 〈φ(v)〉

as φ(〈v〉). This projective version is then not defined on the points of E6(K), and it induces an

involutive bijection from the set of black points onto itself.

In this section, let U be a subspace of P26 entirely consisting of grey points; we will briefly call this

a grey subspace. Then φ(U) corresponds to a set of points of E6(K). We prove some properties of

φ(U).

Lemma 5.1. Let p, q ∈ U, p 6= q. Denote the line joining p and q by L, and note that L ⊆ U.

Then

(i) If φ(p) = φ(q), then φ(L) = φ(p);

(ii) if φ(p) and φ(q) are collinear on E6(K), then φ is bijective on L and φ(L) is a conic on E6(K)

which is contained in a singular plane of E6(K);

(iii) if φ(p) and φ(q) are not collinear on E6(K), then φ is bijective on L and φ(L) is a conic on

E6(K) which is not contained in a singular plane of E6(K).

Proof. Define the cross product v×w as the linearization of φ, i.e., v×w = φ(v+w)−φ(v)−φ(w).

Denote the projective version also by ×, i.e., 〈v〉 × 〈w〉 = 〈v × w〉. Then one calculates that, for

all λ, µ ∈ K,

φ(λv + µw) = λ2φ(v) + λµ(v × w) + µ2φ(w). (5.1)



320 H. V. Maldeghem & M. Victoor CUBO
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If v × w is linearly dependent on φ(v) and φ(w), then also φ(v + w) is a linear combination of

φ(v) and φ(w). Suppose first that φ(v) and φ(w) are not collinear on E6(K). Then, since the only

points of E6(K) on the line 〈v, w〉 are 〈v〉 and 〈w〉, and since 〈φ(v + w)〉 is by assumption a point

of E6(K), we deduce without loss of generality φ(〈v + w〉) = φ(〈v〉). Hence 〈v〉 and 〈v + w〉 are

contained in the same host space, implying 〈w〉 is also, and we are in Situation (i), a contradiction.

Hence Equation (5.1) defines a conic.

Now suppose that φ(v) and φ(w) are collinear on E6(K). Let ξ and ζ be the symplecta the host

spaces of which contain 〈v〉 and 〈w〉, respectively. Let U = ξ∩ζ. Select maximal singular subspaces

V ⊆ ξ and W ⊆ ζ disjoint from U. Then simple dimension arguments show that every point of V

is collinear to a unique point of W . Moreover 〈V, W〉 ∩ E6(K) is a Segre variety S isomorphic to

S1,4(K), and every 4-dimensional generator of that Segre variety is contained in a unique symp

also containing U. This follows from the similar but easy to check fact for S2,2(K) and the fact

that S2,2(K) is amply contained in E6(K) (by [11]). Now v = v1 + v2, with 〈v1〉 ∈ U and v2 ∈ V ,

and w = w1+w2, with 〈w1〉 ∈ U and 〈w2〉 ∈ W . Notice that p×q = φ(p+q) for points p, q ∈ E6(K).

If 〈v2〉 and 〈w2〉 are not contained in the same 1-dimensional generator of S , then 〈v2 + w2〉 is not

contained in S and hence φ(v2 + w2) is not contained in a symplecton through U (as each host

space through U intersects 〈S 〉 in a 4-dimensional generator of S ). Consequently in that case,

v × w = v1 × w2 + v2 × w1 + v2 × w2 = φ(v1 + w2) + φ(v2 + w1) + φ(v2 + w2)

is linearly independent from φ(v) and φ(w) (since φ(v2 + w1) is a (possibly trivial) multiple of

φ(〈v〉) and φ(v1 + w2) a multiple of φ(w)). So in this case, (ii) holds.

So we may assume that 〈v2〉 and 〈w2〉 are collinear on E6(K), i.e., v2 × w2 = ~o. It then follows that

there exists a unique point p on the line through 〈v1〉 and 〈w1〉 collinear with both 〈v2〉 and 〈w2〉 (if

some point q on that line were collinear to 〈v2〉 but not to 〈w2〉, then ζ would be determined by 〈w2〉

and q and would contain 〈v2〉). Hence there exists ℓ ∈ K
× with v2×(v1+ℓw1) = ~o = w2×(v1+ℓw1).

Then, using the bilinearity of the cross-product, we calculate

v × w = v1 × w2 + v2 × w1 = −ℓw1 × w2 − ℓ
−1v1 × v2 = −ℓφ(w) − ℓ

−1φ(v).

Hence, substituting this in Equation (5.1), we obtain

φ(λv + µw) = (λ2 − ℓ−1λµ)φ(v) + (µ2 − ℓλµ)φ(w). (5.2)

which becomes ~o for µ = ℓλ, a contradiction.

The lemma now follows. ✷

We call lines of type (i) short, lines of type (ii) flat and lines of type (iii) conical. We now have

the following result.



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On Severi varieties as intersections of a minimum number of quadrics 321

Proposition 5.2. Set d = dim(U), where we use projective dimensions.

(1) If all lines of U are short, then φ(U) is a point.

(2) If only all lines in a hyperplane of U are short, then either all other lines are flat, or all other

lines are conical. In both cases φ(U) is a quadric of Witt index 1 spanning a (d+1)-dimensional

space in P26, which is singular in the flat case, and in the conical case the quadric is contained

in a symp as a subquadric.

(3) In all cases φ(U) is the quotient (or projection) of a Veronese variety Vd(K), where the image of

a conic is either a conic, or a single point. If U does not contain short lines, then dim〈φ(U)〉 ≥

d.

(4) If U contains two disjoint planes containing only short lines, then every line intersecting both

planes is conical.

For G2,6(K), the last statement becomes:

(4′) If U contains two disjoint short lines, then every line intersecting both lines is conical.

Proof. We start with noting that (1) is obvious: all points of U are contained in the same host

space.

Let e0, . . . , ed be a (vector) basis of U. Then, using the definition of the cross product and the

bilinearity of it, we calculate that φ(U) is the image of the map

(λ0, . . . , λd) 7→
d

∑

i=0

λ2i φ(ei) +

d−1
∑

i=0

d
∑

j=i+1

λiλj(ei × ej), (5.3)

which is a Veronese variety Vd(K) if all φ(ei) and ei × ej are linearly independent. But if not, then

this is just an obvious quotient of Vd(K). If φ(U) does not contain short lines, then no point of the

subspace from which one projects lies on a tangent, and since tangents at one point fill the whole

tangent space, the latter are isomorphically projected. Hence (3).

To show (2), we may assume that all lines of the subspace H := 〈e1, . . . , ed〉 are short. Hence there

exist constants k1, . . . , kd−1 such that φ(ei) = kiφ(ed), ki ∈ K, i = 1, . . . , d−1. Then φ(ei +ej) is a

multiple of φ(ed), i, j ∈ {1, . . . , d}, i 6= j, and so we may write ei × ej = ℓijφ(ed), i, j ∈ {1, . . . , d},

i < j, for some ℓ ∈ K. The mapping (5.3) becomes

(λ0, . . . , λd) 7→ λ
2
0φ(e0) +





d
∑

i=1

kiλ
2
i +

d−1
∑

i=1

d
∑

j=i+1

ℓijλiλj



 φ(ed) +

d
∑

i=1

λ0λi(e0 × ei). (5.4)

If φ(e0), φ(ed) and all e0 × ei, i = 1, . . . , d, are linearly independent from each other, then, with

respect to that basis, and denoting the coordinate corresponding to e0 × ei by Xi, the one corre-

sponding to φ(e0) by X0 and the one corresponding to φ(ed) by Xd+1, it is an elementary exercise to



322 H. V. Maldeghem & M. Victoor CUBO
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calculate that a point is in the image of the map (5.4) if and only if its coordinates (X0, . . . , Xd+1)

satisfy

X0Xd+1 =
d

∑

i=1

kiX
2
i +

d−1
∑

i=1

d
∑

j=i+1

ℓijXiXj. (5.5)

Note that the right hand side of Equation (5.5) is an anisotropic quadratic form; indeed, suppose

there exist xi ∈ K, i = 1, . . . , d, such that
∑d

i=1 kix
2
i +

∑d−1
i=1

∑d
j=i+1 ℓijxixj = 0. Then setting

λ0 = 0 and λi = xi, we see that the right hand side of the map in (5.4) becomes ~o, a contradiction,

as this would yield a white point in U.

Hence Equation (5.5) defines a quadric Q of Witt index 1. If φ(e0), φ(ed) and all e0×ei, i = 1, . . . , d,

are not linearly independent from each other, then φ(S) is a projection of Q. However, considering a

point p in 〈Q〉 in the subspace from which we project, we can select a plane α through p containing

two points of φ(U), and then α contains a conic, which is either not projected bijectively, or

projected into a line, both of which are contradictions to Lemma 5.1. Hence φ(U) spans a space

of dimension d + 1.

If some line L of U is flat, then, for each point p ∈ L\H, φ(p) and φ(L∩H) are collinear on E6(K).

But φ(L ∩ H) = φ(H) = φ(q), for each q ∈ H. Hence all lines of U intersecting L in some point

not in H are flat. Replacing L with each such a line, we obtain that all lines of U not contained

in H are flat.

This completes the proof of (2). We now address (4). Suppose that α and β are two disjoint planes

all of whose lines are short, and suppose for a contradiction that there is a flat line L intersecting

α and β in some point 〈v〉 and 〈w〉, respectively. Then φ(α) = φ(〈v〉) and φ(β) = φ(〈w〉) are

collinear on E6(K). We now use the same notation as in the proof of Lemma 5.1 (ii). So ξ and

ζ are the symplecta with α ⊆ 〈ξ〉 and β ⊆ 〈ζ〉, and U = ξ ∩ ζ. Also, V and W are maximal

singular subspaces of ξ and ζ, respectively, disjoint from U. Let α2 and β2 be the projection of α

and β, respectively, from U onto V and W , respectively. Since 〈V, W〉 ∩ E6(K) is a Segre variety, a

dimension argument implies that some point 〈v2〉 of α2 is collinear on E6(K) with some point 〈w2〉

of β2. But, as one can read in the last part of the proof of Lemma 5.1, this leads to a contradiction.

✷

Corollary 5.3. With the above notation, if U intersects the space spanned by a symp ξ in a

subspace of dimension 1, 2 or 4 in the cases V = S2,2(K), G2,6(K) or E6(K), respectively, then

either U is contained in 〈ξ〉, or V = G2,6(K) and all lines of U that intersect 〈ξ〉 are flat.

Proof. Set d = dim U. Without loss of generality, we may assume that U ∩ 〈ξ〉 is a hyperplane of

U. Then Proposition 5.2 implies that φ(U) is contained in a subspace W of P6d−4 of dimension

d + 1 = 3, 4, 6 for the respective cases. So W can only be a singular subspace of V if V = G2,6(K).

If W is not singular, then φ(U) is quadric of Witt index 1 arising as the intersection of a symp



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On Severi varieties as intersections of a minimum number of quadrics 323

with a subspace of dimension 3, 4 and 6, respectively. But such subspaces always have lines in

common with the symp, since they intersect each maximal singular subspace of the symp in a line,

by an obvious dimension argument, a contradiction. ✷

Now we consider the separate varieties in turn. Notice first that, since V2(K) ⊆ S2,2(K) ⊆

G2,6(K) ⊆ E6(K), each example for a certain variety carries over to the next variety, as ordered in

the inclusions just given.

5.2 The quadric Veronesean V2(K)

Recall that V2(K) is given by the image of the Veronese map

P
2 → P5 : (x, y, z) 7→ (x2, y2, z2, yz, zx, xy).

The line given by the points (0, 0, 0, k, ℓ, 0) entirely consists of grey points, hence in general, 6 −

2 = 4 quadrics suffice to describe V2(K). After a little calculation, ordering the coordinates like

(X1, X2, X3, X23, X31, X12), these turn out to be X1X2 = X
2
12, X3X1 = X

2
31, X2X3 = X

2
23, and

any one of X1X23 = X31X12, X2X31 = X12X23 or X3X12 = X23X31. In characteristic 2, the

whole nucleus plane consists of grey points and hence the first three equations suffice (see also

Lemma 4.20 in Hirschfeld & Thas [6]). This somehow reflects the property of the gradient being

identically zero in the last three coordinates.

We now determine all grey planes, showing in particular that in characteristic not equal to 2 there

do not exist such planes, and in characteristic 2 only the nucleus plane is a grey plane, except if

the underling field is F2. We are grateful to J. Thas for hinting the use of conic bundles in the

below argument (our original proof consisted merely of boring calculations).

So suppose π is a grey plane containing at least one point p contained in a secant L. Obviously there

are no flat lines. Then Corollary 5.3 implies that π only contains conical lines. Proposition 5.2(3)

now implies that φ is bijective from π onto V2(K). Hence the map ρ mapping each point p ∈ π

to the unique conic C on V2(K) with p ∈ 〈C〉 is a bijection. Let L ∩ V2(K) = {x, y}. Consider

the bundle B of conics of P2 defined by intersecting V2(K) with all hyperplanes containing the

solid 〈π, L〉. By the bijectivity of ρ, each conic D on V2(K) containing x generates, together with

π and L, a hyperplane HD. Hence HD ∩ V2(K) is a degenerate conic in P
2, which also contains

y. So if y /∈ D, then HD ∩ V2(K) contains a conic of V2(K) through y. It follows that B consists

solely of degenerate conics. But an arbitrary pair of members of B not containing the line of P2

corresponding to the conic of V2(K) containing x and y generates a bundle containing exactly three

degenerate members. Hence |K| = 2. In this case one can easily check that π is the unique plane

in a solid spanned by the complement in V2(F2) of a conic (a conic corresponding to a line of P
2
F2
).

Hence there are seven such planes. Each such plane intersects the nucleus plane in a unique point,



324 H. V. Maldeghem & M. Victoor CUBO
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namely the unique point of the solid not lying in a plane spanned by any three of the four points

of V2(F2) it contains.

5.3 The Segre variety S2,2(K)

This is the only case with a uniform answer for arbitrary fields. Indeed, we will show that there

always exists a grey plane π, and never a grey solid.

First, if we represent S2,2(K) as the 3 × 3 rank 1 matrices, up to a scalar, then we can define π as

the plane containing all skew-symmetric matrices (with 0 on every diagonal entry). It is easy to see

that a skew-symmetric matrix, which always has determinant 0, has rank 1 if and only if it is the

0-matrix. Here every line of π is conical and φ(π) is a Veronesean isomorphic to V2(K) embedded

in S2,2(K). It follows that the following system of equations in the unknowns X00, . . . , X22 defines

S2,2(K):














































X11X22 = X12X21

X00X22 = X02X20

X00X11 = X01X10

X10X02 + X01X20 = (X12 + X21)X00

X01X12 + X10X21 = (X02 + X20)X11

X02X21 + X20X12 = (X01 + X10)X22

In characteristic 2, the plane π is the nucleus plane of the Veronese surface contained in S2,2(K)

obtained by restricting S2,2(K) to the symmetric (rank 1) 3 × 3 matrices.

Now suppose there exists a grey solid S. If S contains a short line, then considering any plane

in S containing that short line, Corollary 5.3 leads to a contradiction. If S contains only conical

lines, then let L1 and L2 be two non-intersecting lines of S. Let ξi be the symp containing φ(Li),

i = 1, 2. Clearly ξ1 6= ξ2 as otherwise every point of φ(L1) is collinear to two points of φ(L2),

yielding flat lines. Hence ξ1 and ξ2 intersect nontrivially and since φ(L1) is an ovoid of ξ1, some

point x1 ∈ φ(L1) is collinear to a point of the intersection ξ1 ∩ ξ2. Then x1 is collinear to a line of

ξ2, and since φ(L2) is an ovoid of ξ2, x1 is collinear to some point x2 ∈ φ(L2), a contradiction (as

〈x1, x2〉 is then the image under φ of a flat line of S). Hence there is at least one flat line L ⊆ S. Let

π be the plane spanned by φ(L). If φ(S) ⊆ π, then S only contains flat lines. By Lemma 5.2 (3) the

dimension of π is at least 3, a contradiction. Hence there is some point p ∈ S with φ(p) /∈ π. Since

there is a unique point in π collinear to φ(p), we can pick two points x1, x2 ∈ L such that φ(xi) is

not collinear to φ(p), i = 1, 2. Let ξi be the symp determined by φ(xi) and φ(p), i = 1, 2. Then

ξ1 ∩ ξ2 is obviously equal to the line through φ(p) intersecting π. The argument above shows that

for each point q1 on the line 〈x1, p〉, the point φ(q1) is collinear in S2,2(K) to a unique point φ(q2),

with q2 ∈ 〈x2, p〉. But clearly φ(〈q1, q2〉) is contained in a plane disjoint from π, contradicting the

fact that 〈p, L〉 is a projective plane in S. So we ruled out all possibilities for S to exist.



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On Severi varieties as intersections of a minimum number of quadrics 325

Nevertheless one can sometimes find other grey planes. For instance, if P2 admits a linear

collineation without fixed points, then one can find such a plane in the span of two disjoint singular

planes of S2,2(K). Such a plane only has flat lines. As an example suppose K is a field admitting

a cubic extension L; let the corresponding cubic polynomial be given by x3 − T x2 + Qx − N, with

T, Q, N ∈ K. The plane containing the points









m k − Qm −ℓ

ℓ − T m Nm k

0 0 0









, k, ℓ, m ∈ K,

is grey, as one can calculate (in the calculations one might need the fact that also the polynomial

x3 + Qx2 + T Nx + N2 is irreducible; its roots in the cubic extension L are the opposites of the

pairwise products of the roots of the original polynomial). Applying φ we obtain that the mapping

(k, l, m) 7→









0 0 0

0 0 0

k2 + Nℓm − Qkm ℓ2 − T ℓm + km (N − QT )m2 − lℓ + T km + Qℓm









,

k, ℓ, m ∈ K, induces a bijection from P2 onto a singular plane of S2,2(K), where each line is mapped

to a conic. In fact, these conics form the net of all conics passing through three given conjugate

points in the plane over the cubic extension L.

Remark 5.4. One might wonder how the net of conics in P2 of the last example can be a projection

of the quadric Veronese surface, as required by Proposition 5.2 (3). To see this directly, one

considers the above net of conics in P2, take its image under the Veronese map, and project the

Veronese surface from the intersection of the hyperplanes spanned by the image of three linearly

independent members of the net. This intersection is a plane consisting merely of black points.

5.4 The line Grassmannian G2,6(K)

By the previous subsection, there always exists a grey plane. But we can do better for certain

fields, in particular, if the field K admits a quadratic extension (separable or not). We will see

that in this case we can find a grey 5-dimensional subspace of P14. But we start with a curious

example in the case that P3 admits a linear collineation without fixed elements.

Example 5.5 (Dimension 3). Therefore, we consider a point p ∈ G2,6(K) and the subspace Up ⊆

P
14 generated by all singular lines on p. Then dim Up = 8 and G2,6(K) ∩ Up is a cone with vertex

p and base S1,3(K) (the latter is indeed the residue at p). Consider any base space W; that is, a

7-dimensional subspace of Up not containing p. Then S := W ∩ G2,6(K) ∼= S1,3(K). Take two

singular solids S1, S2 of S . The mapping θ : S1 → S2 : x1 7→ x2 defined by 〈x1, x2〉 ⊆ S is a

(linear) collineation from S1 to S2. Now let ϕ be a linear collineation of S2 without fixed elements.



326 H. V. Maldeghem & M. Victoor CUBO
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Then let S ′ ∼= S1,3(K) be the Segre variety with as set of maximal singular 1-dimensional subspaces

the set of lines {〈p, pθϕ〉}, and select a 3-dimensional singular subspace S′ of S ′ distinct from S1

and S2. We claim that S
′ ∩ S = ∅. Indeed, each point in S′ is on a unique line intersecting both

S1 and S2, and if that line would belong to S and intersect Si in xi, i = 1, 2, then x
θ
1 = x2 = x

θϕ
1 ,

implying x2 is a fixed point of ϕ, a contradiction.

Suppose now that two points p′, q′ ∈ S′ are contained in a common host space of some symplecton

ξ of G2,6(K). Then it is easy to see that ξ ∩ Si = Li is a line, i = 1, 2. In the solid 〈L1, L2〉 there

is a unique line L′ containing p′ and intersecting Li in some point pi; then p
θϕ
1 = p2. Likewise,

there is a unique line M′ containing q′ and intersecting Li in some point qi; then q
θϕ
1 = q2. Hence

L
θϕ
1 = 〈p1, q1〉

θϕ = 〈p2, q2〉 = L2. But the latter also coincides with L
θ
1 (as L1, L2 is contained in

a hyperbolic quadric completely contained in S ). Hence ϕ fixes L2, a contradiction.

We conclude that φ(S′) is the bijective projection of a Veronese variety V3(K) into a hyperbolic

quadric in some 5-dimensional projective space (that quadric corresponds to the point p; it consists

of the images under φ of the symplecta passing through p). This is a rather remarkable situation.

But that inclusion can abstractly be seen directly by sending a point x of S2 to the image of the line

〈x, xϕ〉 under the Klein correspondence. We deduce that every plane of the Klein quadric contains

a unique conic of that image.

If, in the above, ϕ has no fixed points, but does admit fixed lines, then we can still find S′ and

it is still a grey solid. But φ(S′) is the union of elliptic quadratic surfaces (in 3-dimensional

subspaces). An extreme situation is that the fixed lines of ϕ form a spread of S′, in which case

φ(S′) coincides with one such elliptic quadric. It is clear that this situation arises if and only if K

admits a quadratic extension. But in this case we can extend S′ to a 5-dimensional grey subspace,

as evidenced by the next example.

Example 5.6 (Dimension 5). Let x2 − T x + N be an irreducible quadratic polynomial over K

(with coefficients in K), defining the quadratic extension L of K. Let p1, p2, p3 be three points on

a line of the quadrangle Γ′ of order (2, 2), and suppose {p1, p2, p3} is a spread line. Let {pi, qi, ri}

and {pi, si, ti} be the other two lines passing through pi, i = 1, 2, 3. We may choose this notation

such that {q1, q2, q3} and {r1, r2, r3} are the other two spread lines in Γ
′, and the other six lines of

Γ′ are {s1, q2, t3}, {s1, t2, r3} and cyclic permutations of the indices. (For an explicit realization

inside the model given in Subsection 2.2, see Example 5.8.) Define the following subspace:















0 = Xpi, i = 1, 2, 3,

0 = Xri + Xqi, i = 1, 2, 3,

0 = Xti + NXsi + T Xqi, i = 1, 2, 3.

Since we have nine linearly independent equations, this defines a 5-dimensional projective subspace

U. In order to apply φ we write a generic point of U with coordinates Xpi = 0, i = 1, 2, 3, the

coordinates Xqi and Xsi are considered as running parameters, i = 1, 2, 3, and the coordinates



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On Severi varieties as intersections of a minimum number of quadrics 327

Xri and Xti linearly depend on these parameters as given above, namely Xri = −Xqi and Xti =

−NXsi−T Xqi. We denote the coordinate vector of such a generic point by vXq1 ,Xq2 ,Xq3 ;Xs1 ,Xs2 ,Xs3 ,

or simply v in the sequel. Calculating φ(v) we obtain a vector with pi-coordinate equal to

X2qi + T Xq1Xsi + NX
2
si
, i = 1, 2, 3.

Clearly, such coordinate is 0 if and only if Xqi = Xsi = 0, showing that no point of U is white.

Calculating C(v), we simply obtain 0, showing that U is a grey space.

One now sees that the short lines in U form a regular spread; they are the point set of a projective

plane P2
L
the lines of which are the 3-dimensional subspaces of U generated by two distinct short

lines. This is the spread representation of P2
L
. We now claim that φ transforms this representation

into the corresponding Hermitian Veronesean of P2
L
. Indeed, let δ be one of the roots in L of the

polynomial x2 − T x + N, and let x 7→ x be the corresponding Galois involution of L. Note that

a + bδ = a + T b − bδ, a, b ∈ K. Denoting the p-coordinate of φ(v) by Yp, p ∈ P, a straightforward

calculation reveals:

(Xq2 + Xs2δ)(Xq3 + Xs3δ) = Yr1 + Yt1δ,

(Xq1 + Xs1δ)(Xq1 + Xs1δ) = Yp1,

Ys1 = NYt1,

Yq1 = Yr1 − T Yt1,

and the same equation for cyclic permutations of the indices, which shows that φ(U) is projectively

equivalent to the point set

{(X1X1, X2X2, X3X3, X2X3, X3X1, X1X2) | X1, X2, X3 ∈ L},

where the first three coordinates are considered to belong to K, and the last three to K × K via the

obvious identification a + bδ → (a, b). This shows our claim.

5.5 The Cartan variety E6(K)

By the previous subsections, there always exists a grey plane, and if K admits a quadratic extension,

there is always a grey 5-space. We can slightly generalise the latter, and we can also give an example

of a grey 11-dimensional space if K is the centre of a quaternion division algebra, or char K = 2

and K admits a degree 4 inseparable field extension. Also, we will show that there always exists a

grey 4-space, whatever the field.

Example 5.7 (Dimensions 4 and 5). Let Γ′ = (P′, L ′) be a subquadrangle of Γ = (P, L ) of order

(2, 2). Let W be the 12-dimensional vector subspace of K27 generated by the ep not belonging to Γ
′.

The points outside P′ form a double six {p1, . . . , p6, q1, . . . , q6}, where {p1, . . . , p6} and {q1, . . . , q6}



328 H. V. Maldeghem & M. Victoor CUBO
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are cocliques and pi is collinear to qj if and only if i 6= j, for all i, j ∈ {1, . . . , 6}. Let w ∈ W have

coordinates (xp)p∈P (with xp = 0 if p ∈ P
′). Then φ(w) = 0 if and only if xpixqj = ±xpj xqi,

for all i, j ∈ {1, . . . , 6} with i 6= j, and where each sign depends on the position of the spread S .

However, changing the sign of the coordinates related to the points of one single six collinear to

the points of one single three with respect to the grid in Γ′ defined by intersecting L ′ with S ,

we see that all signs become positive. This means that, denoting the projective subspace defined

by W also by W, the intersection W ∩ E6(K) is a Segre variety S1,5(K). As before, given a fixed

point free linear collineation of P5, one can select a 5-dimensional subspace U of W disjoint from

E6(K), which is automatically a grey subspace. If K admits a separable quadratic extension, then

we may choose W such that it contains a regular spread of short lines, and φ(U) is a Hermitian

Veronesean variety on E6(K), as in Example 5.6. However, note that in the inseparable case, the

corresponding spread is elementwise fixed only by the identity. We hence conjecture that also in

the separable case, the current 5-space is not projectively equivalent to the one of Example 5.6

(meaning the current subspace U is not contained in the space spanned by any subvariety of E6(K)

isomorphic to G2,6(K)).

Now let K be arbitrary and let M be a 6 × 6 upper triangular matrix with entries in K, with 1s on

the diagonal and such that M − I (with I the identity matrix) has rank 5. Then the corresponding

linear collineation θ of P5 has a unique fixed point. Let U′ be a 5-space in W constructed as above

from θ; then U′ ∩ E6(K) is a point p corresponding to the unique fixed point of θ. Hence any

hyperplane of U′ not containing p is a grey 4-space.

Example 5.8 (Dimension 11). Let x21 − T x1x2 + Nx
2
2 − ℓx

2
3 + ℓT x3x4 − ℓNx

2
4 be the norm form

of a quaternion division algebra H over K, with ℓ, T, N ∈ K, or with T = 0 and char K = 2, and

then we assume it is just an inseparable field extension of degree 4.

It is convenient to work with the explicit description of Γ and S given in Subsection 2.2. The

current example will extend Example 5.6 with

(p1, p2, p3) = (25, 14, 36),

(q1, q2, q3) = (34, 26, 15),

(r1, r2, r3) = (16, 35, 24),

(s1, s2, s3) = (46, 56, 45),

(t1, t2, t3) = (13, 23, 12).

The subspace U we want to define can be described by a system of fifteen equations, nine of which are

given in Example 5.6 (using the above identification). The other six read (denoting the coordinate

corresponding to the point i by Xi and the one corresponding to i
′ by X′i, i = 1, . . . , 6):







0 = X′i − ℓXj, (i, j) = (2, 5), (1, 4), (3, 6),

0 = X′i − ℓNXj − ℓT Xi, (i, j) = (5, 2), (4, 1), (6, 3).



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On Severi varieties as intersections of a minimum number of quadrics 329

In completely the same way as in Example 5.6, one checks that U is a grey 11-dimensional subspace,

and that its image under φ is the corresponding quaternion Veronesean of P2
H
.

As a corollary of the last example, we obtain that every quaternion Veronese variety of the plane

P
2
H
, with H a quaternion division algebra over the field K, or a degree 4 inseparable field extension

in characteristic 2, is a projection of the Veronese variety V11(K).

Over the real numbers, we can choose −ℓ = N = 1 and T = 0. It follows that in this case E6(R)

has a particularly nice description as the intersection of fifteen quadrics, whose forms can be given

as follows. Choose a fixed spread line L of S . Three of the forms are Qp, with p ∈ L. The other

twelve forms are all of shape Qa + Qb, where {a, b, p} ∈ L is a line of Γ with p ∈ L.

5.6 Conclusion

We conclude by noting that we gave a full answer for the minimality of the number of quadrics

describing a Severi variety in the cases of V2(K) and S2,2(K). For the two other case, we were only

able to give some examples (yielding bounds) over fields with certain properties. Since we think

that some of the dimensions we obtained are pretty high, we conjecture that

(C1) If K admits a quadratic extension, then the maximum projective dimension of a grey subspace

for G2,6(K) is 5.

(C2) If K admits a quaternion division algebra, or a degree 4 inseparable field extension in char-

acteristic 2, then the maximum projective dimension of a grey subspace for E6(K) is 11.

Remark 5.9. We note that the minimum number of quadrics found in the present paper for

a certain variety, is exactly equal to the dimension of the vector space related to the variety of

the previous case, ranking the cases in increasing dimension, and adding a trivial variety in the

beginning consisting of three spanning points in a projective plane (three 1-spaces generating a

3-dimensional vector space; this is the line-residue of the long root geometry of type D4 which is

sometimes added as zeroth column in the fourth row of the Freudenthal-Tits magic square; the

Severi varieties are the line-residues of the other varieties of the fourth column). We do not think

this is a coincidence; further research should give evidence for this.

Finally, one could wonder which quadrics one can obtain by linearly combining the 27 basic quadrics

in the case E6(K), or 9 and 15 basic quadrics in the cases S2,2(K) and G2,6(K), respectively.

It is proved in a yet unpublished manuscript of A. De Schepper and M. Victoor that there are

exactly three possibilities (corresponding to the “duals” of the white, grey and black points): For

E6(K), these are non-degenerate parabolic quadrics (hence of maximal Witt index) and degenerate

quadrics with an 8- or 16-dimensional radical (projective dimension) and hyperbolic base. Similarly,



330 H. V. Maldeghem & M. Victoor CUBO
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for G2,6(K), we have non-degenerate parabolic quadrics and degenerate quadrics with a 4- or 8-

dimensional radical and hyperbolic base; for S2,2(K), we have non-degenerate parabolic quadrics

and degenerate quadrics with a 2- or 4-dimensional radical and hyperbolic base.



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On Severi varieties as intersections of a minimum number of quadrics 331

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	Introduction
	Preliminaries
	The varieties
	A generalized quadrangle

	Main result
	Proof of Theorem 3.1
	A cubic form
	A lemma
	Reducing the number of quadrics|End of the proof

	Examples and applications
	General observations
	The quadric Veronesean V2(K)
	The Segre variety S2,2(K)
	The line Grassmannian G2,6(K)
	The Cartan variety E6(K)
	Conclusion