On invariant notions of Segre varieties in binary projective spaces

On invariant notions of Segre varieties in binary projective spaces

Hans Havlicek    Boris Odehnal    Metod Saniga
Abstract

Invariant notions of a class of Segre varieties of that are direct products of copies of PG, being any positive integer, are established and studied. We first demonstrate that there exists a hyperbolic quadric that contains and is invariant under its projective stabiliser group . By embedding PG into , a basis of the latter space is constructed that is invariant under as well. Such a basis can be split into two subsets whose spans are either real or complex-conjugate subspaces according as is even or odd. In the latter case, these spans can, in addition, be viewed as indicator sets of a -invariant geometric spread of lines of . This spread is also related with a -invariant non-singular Hermitian variety.

The case is examined in detail to illustrate the theory. Here, the lines of the invariant spread are found to fall into four distinct orbits under , while the points of form five orbits.

Mathematics Subject Classification (2010): 51E20, 05B25, 15A69
Key words: Segre variety, binary projective space

1 Introduction

The present note is concerned with invariant notions in the ambient space of certain Segre varieties over fields of characteristic two and, in particular, over the smallest Galois field . The attribute invariant always refers to the stabiliser of the Segre in the projective group of the ambient space.

Our text is organised as follows: In Section 2 we collect some background results about those Segre varieties which are products of projective lines over a field . The next section presents an invariant quadric of a Segre for and a ground field of characteristic two (Theorem 1). This quadric is regular, of maximal Witt index, contains the given Segre, and its polarity is the fundamental polarity of this Segre. The following sections deal with Segre varieties over . By extending the ground field of the ambient space from to we find an invariant basis (Theorem 2) and an invariant Hermitian variety (Section 5). The theory splits according as is even or odd. In the latter case there is an invariant geometric line spread (Corollary 1) which gives also rise to a spread of the invariant quadric. We make use of our previous results in Section 6, where we describe certain line orbits and all point orbits of the stabiliser group of the Segre in terms of the invariant basis. This complements [14, p. 82], where a completely different description of these point orbits was given without proof.

There is a widespread literature on closely related topics, like notions of rank of multi-dimensional arrays [8], secant varieties of Segre varieties (mainly over the complex numbers) [1], the very particular properties of certain Segre varieties over [24], [25], quantum codes [14], and entanglement of quantum bits in physics [10], [15], [19]. The few sources which are cited here contain a wealth of further references.

2 Notation and background results

Let be a commutative field and let be two-dimensional vector spaces over . So are projective lines over for . We consider the tensor product and the projective space . The non-zero decomposable tensors of determine the Segre variety (see [7], [17])

of . This Segre will also be denoted by .

We recall some facts which are well known from the classical case [7, p. 143], where is the field of complex numbers. They can immediately be carried over to our more general settings. Given a basis for each vector space , , the tensors

(1)

constitute a basis of . For any multi-index the opposite multi-index, in symbols , is characterised by for all . In other words, two multi-indices are opposite if, and only if, their Hamming distance is maximal.

Let for . Then

(2)

denotes their Kronecker (tensor) product. Each permutation gives rise to linear bijections sending to . Also, the symmetric group acts on via . There is a unique mapping

(3)

Clearly, this depends on the chosen bases. The subgroup of preserving decomposable tensors is generated by all transformations of the form (2) and (3). It induces the stabiliser of the Segre within the projective group .

Each of the vector spaces admits a symplectic (i. e., non-degenerate and alternating) bilinear form111We use the same symbol for all these forms. Note that , since for all . This coincidence of a symplectic group with a special linear group underpins much of the mathematics used in this article. . Consequently, is equipped with a bilinear form, again denoted as , which is given by

(4)

and extending bilinearly. Like the forms on this bilinear form on is unique up to a non-zero factor in . In projective terms the form on (or any proportional one) determines the fundamental polarity of , i. e., a polarity which sends to its dual. This polarity is orthogonal when is even and , but null otherwise: Indeed, it suffices to consider the tensors of our basis (1). Given we have

(5)
(6)

Hence is symmetric when is even and , and it is alternating otherwise.

Let be even and . Then is a quadratic form having Witt index and rank . So, the fundamental polarity of the Segre is the polarity of a regular quadric. The Segre coincides with this quadric precisely when .

3 The invariant quadric

We now focus on the case when has characteristic two. Here is a symplectic bilinear form on for any , whence the fundamental polarity of the Segre is always null. Furthermore, (5) simplifies to

(7)
Proposition 1.

Let and . Then there is a unique quadratic form satisfying the following two properties:

  1. vanishes for all decomposable tensors.

  2. The symplectic bilinear form is the polar form of .

Proof.

(a) We denote by the set of all multi-indices with . In terms of our basis (1) a quadratic form is given by

(8)

Given an arbitrary decomposable tensor we have

where we used (7), , , and . This verifies property 1.

(b) Let be arbitrary multi-indices. Polarising gives

The numerator of a summand of the above sum can only be different from zero if and . These conditions can only be met for , whence in fact at most one summand, namely for the one with , and for the one with , can be non-zero. So

and, irrespective of whether or , we have

But this implies that the quadratic form polarises to , i. e., also the second property is satisfied.

(c) Let be a quadratic form satisfying properties 1 and 2. Hence the polar form of is zero. We consider as a vector space over its subfield comprising all squares in . So is a semilinear mapping with respect to the field isomorphism ; see, e. g., [9, p. 33]. The kernel of is a subspace of which contains all decomposable tensors and, in particular, our basis (1). Hence vanishes on , and as required. ∎

From (8) and (7), the quadratic form can be written in terms of tensor coordinates as

(9)

The previous results may be slightly simplified by taking symplectic bases, i. e., for all , whence also for all .

Observe also that Proposition 1 fails to hold for . A quadratic form vanishing for all decomposable tensors of is necessarily zero, since any element of is decomposable. Hence the polar form of such a cannot be non-degenerate.

Theorem 1.

Let and . There exists in the ambient space of the Segre a regular quadric with the following properties:

  1. The projective index of is .

  2. is invariant under the group of projective collineations stabilising the Segre .

Proof.

Any , , preserves the symplectic form on to within a non-zero factor. Any linear bijection as in (3) is a symplectic transformation of . Hence any transformation from the group preserves the symplectic form (4) up to a non-zero factor. Consequently, also is invariant up to a non-zero factor under the action of .

From (9) the linear span of the tensors with ranging in is a singular subspace with respect to . So the Witt index of equals , because being non-degenerate implies that a greater value is impossible.

We conclude that the quadric with equation has all the required properties. ∎

We henceforth call the invariant quadric of the Segre . The case deserves special mention, as the Segre coincides with its invariant quadric given by . This result parallels the situation for .

4 The invariant basis

In what follows will denote the Galois field with elements. We adopt the notation and terminology from Section 2, but we restrict ourselves to the case . Indeed, we shall always identify with the prime field of , where . For each we fix a basis so that we obtain the tensor basis (1).

Let be any vector space over . Then can be embedded in , which is a vector space over , by ; see, for example, [18, p. 263]. We shall not distinguish between and . Likewise, if denotes a linear mapping between vector spaces over , then the unique linear extension of to the corresponding vector spaces over will also be written as rather than . Similarly, we use the same symbol for a bilinear form on and its extension to a bilinear form on . After similar identifications, we have , , and so on. We make use of the usual terminology for real and complex spaces also in our setting. We address the vectors of to be real, we speak of complex-conjugate vectors, points, and subspaces. In particular, a subspace is said to be real if it coincides with its complex-conjugate subspace.

Applying this extension to our vector spaces and their tensor product gives vector spaces and . The last vector space can be identified with in a natural way, so that the Segre can be viewed as as subset of . Likewise, we have .

From now on we shall make use of the following observation: When the projective line is embedded in advantage can be taken from the fact that there is a unique projective basis consisting of the complex-conjugate pair of points, while there is a choice of three different pairs of points for a projective basis of .

Theorem 2.

For each let and be the only two points of the projective line that are not contained in . Then

is a basis of which is invariant, as a set, under the stabiliser of the Segre .

Proof.

We may assume that

(10)

As and are linearly independent, the tensors

(11)

constitute a basis of , whence is a projective basis. The invariance of under follows from the fact that the points and are determined uniquely up to relabelling. ∎

We shall refer to as the invariant basis of the Segre . In order to describe the action of the stabiliser group of the Segre on the invariant basis we need a few technical preparations:

First, from now on the set of multi-indices will be identified with the vector space . Secondly, for any -dimensional vector space over we can define the -valued sign function to be if induces an even permutation of and otherwise.

Proposition 2.

The stabiliser group of the Segre has the following properties:

  1. Let for and write

    (12)

    The collineation given by sends any point to the point .

  2. acts transitively on the invariant basis .

  3. Let be a permutation and define as in (3). Then sends any point to .

Proof.

(a) Each mapping induces a projectivity of the projective line which stabilises .

If then the restriction to is an even permutation, namely either a permutation without fixed points or the identity on . In the first case the characteristic polynomial of has two distinct zeros over , whence each of the two points and remains fixed. In the second case all points of are fixed.

If then gives a permutation of with precisely one fixed point. Such an is an involution, whence the points and are interchanged.

We infer from the above results that sends the point of with multi-index to the point of with multi-index .

(b) Given we let . In order find a collineation from taking to , it suffices to choose for all some with . This can clearly be done, so that yields a collineation with the required properties.

(c) According to the (basis-dependent) definition of in (3), we have to consider the linear bijections sending to . By (10), any such map sends to . Now follows immediately. ∎

The parity of a point can be defined as the parity of the multi-index (i. e., it is even or odd according to the number of s among the entries of ). We write and for the set of base points with even and odd parity, respectively. Even though we can distinguish points of even and odd parity due to our fixed bases , a change of bases in the vector spaces may alter the parity of a point. But having the same parity is an equivalence relation on with two equivalence classes, namely and , each of cardinality .

We define the Hamming distance between and as the Hamming distance of their multi-indices and . In particular, we speak of opposite points if and are opposite. For each point of there is a unique opposite point. By (10) and (11) opposite points of are complex-conjugate with respect to the Baer subspace of . The opposite point to can also be characterised as the only point such that . We remark that the Hamming distance on admits another description due to . The Hamming distance of equals the number of lines on a shortest polygonal path in from to .

From the proof of Proposition 2 and the above remarks we immediately obtain:

Theorem 3.

The stabiliser group of the Segre acts on the invariant basis (via the multi-indices of its points) as the group of all affine transformations of having the form , where and . Hence, Hamming distances on are preserved under , and the partition is a -invariant notion.

We now use the invariant basis for describing some other -invariant subsets. In the following theorem we also make use of a particular property of Segre varieties over . Recall that (for an arbitrary ground field ) there are precisely generators through any point of the Segre . They span the (-dimensional) tangent space of at . The tangent lines at are the lines through which lie in its tangent space. For there are tangents at . Precisely one of them does not lie in any of the -dimensional subspaces which are spanned by generators through . This line will be called the distinguished tangent at .

Theorem 4.

The stabiliser group of the Segre has the following properties:

  1. The union of the skew subspaces and is a -invariant subset of .

  2. The union of the mutually skew real lines222Any line joining complex-conjugate points is real (cf. the beginning of Section 4). It carries three real points. We use the symbol for the join of points.

    (13)

    is a -invariant subset. The real points on these lines comprise an orbit of .

  3. If is even then and are real subspaces. Each of the lines from (13) is contained in precisely one of them.

  4. If is odd then and are complex-conjugate subspaces. All lines from (13) meet and at precisely one point, respectively.

  5. All distinguished tangents of the Segre meet and at precisely one point, respectively.

Proof.

Ad 1 and 2: The assertions on the invariance of and on the invariance of the union of all lines from (13) are a direct consequence of Theorem 3.

We denote the set of all real points on the lines from (13) by . Let and let be an arbitrary real point on the line . Any collineation from takes to some real point on a line from (13), whence the orbit of is contained in .

Conversely, let . So there is a with . By Theorem 3, there exists a collineation in which maps to and, consequently, also to . Furthermore, is mapped to some real point on the line . There exists with . Then and are eigenvectors of with eigenvalues and , where . (See the proof of Proposition 2.) The linear bijection has and as eigenvectors with eigenvalues and , respectively, due to . Thus the collineation arising from induces a non-identical even permutation on the three real points of the line . Such a permutation has only one cycle. So one of , or yields a collineation from which maps to .

Ad 3 and 4: Opposite points of are complex-conjugate and vice versa. Such points share the same parity for even, but have different parity for odd.

Ad 5: First, we exhibit the distinguished tangent of the Segre at the point . On each of the generators of the Segre through this point we select one more real point, namely , , …, for facilitating our subsequent reasoning. So, the distinguished tangent contains the real point

(14)

By (10), we have and for all . So and

Mutatis mutandis, we obtain linear combinations for with coefficients . Summing up gives

(15)

where

There are two cases:

even: Due to we have for all with even parity and for all with odd parity. So meets at the point (14). The sum of the tensor from (15) and determines the point of intersection of with .

odd: Due to we have for all with even parity and for all