General Remarks 1 Finite Classical Polar Spaces 2 m -ovoids and x - - PowerPoint PPT Presentation

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

Tight Sets and m-Ovoids of Quadrics 1

Qing Xiang

Department of Mathematical Sciences University of Delaware Newark, DE 19716 USA xiang@math.udel.edu

Joint work with Tao Feng and Koji Momihara

• 1T. Feng, K. Momihara, Q. Xiang, Cameron-Liebler line classes with

parameters x = q2−1

2

, J. Combin. Theory (A), 133 (2015), 307–338

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

1

General Remarks

2

Finite Classical Polar Spaces

3

m-ovoids and x-tight sets

4

Cameron-Liebler line classes

5

m-Ovoids of Q(4, q)

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

General Remarks

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

Substructures in Projective and Polar Spaces

This is a talk about substructures in projective and polar spaces, such as arcs, ovals, ovoids, tight sets, spreads. These objects are not only interesting in their own right, but also can give rise to

• ther combinatorial objects, such as translation planes, designs,

strongly regular graphs, two-weight codes, and association schemes, etc.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

A sample result

Let V be an (n + 1)-dimensional vector space over Fq. We will consider PG(n, q), whose points are the 1-dimensional subspaces

• f V , lines are the 2-dimensional subspaces of V , planes are the

3-dimensional subspaces of V , and so on. A subset A of points of PG(2, q) is called an arc if no three points

• f A are collinear (in other words, every line meets A in 0, 1, or 2

points); one can show that |A| ≤ q + 2, and furthermore |A| ≤ q + 1 if q is odd. Arcs of size q + 1 are called ovals.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

Segre’s Theorem

Theorem (Segre, 1955). In PG(2, q) where q is odd, every oval is a non-singular conic. In other words, every oval is projectively equivalent to the conic Y 2 = XZ, which is the set of points {(1, t, t2) | t ∈ Fq} ∪ {(0, 0, 1)}. In contrast, ovals (and hyperovals) in PG(2, q), q even, are far from being classified.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

Finite Classical Polar Spaces

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

Definitions

Let V (n + 1, q) be an (n + 1)-dimensional vector space over Fq, and let f be a non-degenerate sesquilinear or non-singular quadratic form defined on V (n + 1, q). A finite classical polar space associated with the form f is the geometry consisting of subspaces of PG(n, q) induced by the totally isotropic subspaces with relation to f.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

A polar space S contains the totally isotropic points, lines, planes,

• etc. of the ambient projective space.

The generators of S are the (t.i.) subspaces of maximal dimension. The rank of S is the vector dimension of its generators For a point P, the set P ⊥ of points of S collinear with P is the intersection of the tangent hyperplane at P with S.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

Three types of finite classical polar spaces: Orthogonal polar spaces: quadrics; symplectic polar spaces; Hermitian polar spaces. Polar Space rank form Q(2n, q) n x2

0 + x1x2 + · · · + x2n−1x2n

Q+(2n + 1, q) n + 1 x0x1 + x2x3 + · · · + x2nx2n+1 Q−(2n + 1, q) n f(x0, x1) + x2x3 + · · · + x2nx2n+1 W(2n + 1, q) n + 1 x0y1 + y0x1 + · · · + x2ny2n+1 + x2n+1y2n H(2n, q2) n xq+1 + · · · + xq+1

2n

H(2n + 1, q2) n + 1 xq+1 + · · · + xq+1

2n+1

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

m-ovoids and x-tight sets

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

Let S be a finite classical polar space of rank r over the finite field

• Fq. Denote by θn(q) := qn−1

q−1 the number of points in

PG(n − 1, q). Definition An m-ovoid is a set O of points such that every generator of S meets O in exactly m points. Definition An x-tight set is a set M of points such that |P ⊥ ∩ M| =

• xθr−1(q) + qr−1,

if P ∈ M, xθr−1(q),

• therwise.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

Example A spread of PG(3, q) is mapped, under the Klein correspondence, to an ovoid of the Klein quadric Q+(5, q). (A spread of PG(3, q) is a set of q2 + 1 lines partitioning the set of points of PG(3, q).) Ovoids of polar spaces are rare: they only exist in low rank polar spaces, such as Q(4, q), Q(6, q), Q+(5, q), Q+(7, q). Example Let S be a polar space of rank r. Then any generator M is a 1-tight set of S since |P ⊥ ∩ M| =

• |M| = qr−1

q−1 = θr−1(q) + qr−1,

if P ∈ M, θr−1(q),

• therwise.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

Cameron-Liebler Line Classes

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

Background

Cameron-Liebler line classes were first introduced by Cameron and Liebler 2 in their study of collineation groups of PG(n, q), n ≥ 3, having the same number of orbits on points as on lines. Cameron and Liebler reduced the problem to the case where n = 3.

2P.J. Cameron, R.A. Liebler, Tactical decompositions and orbits of projective groups, Linear Algebra Appl., 46 (1982), 91–102.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

A collineation group of PG(3, q) having equally many orbits on points and lines induces a symmetric tactical decomposition on the point-line design from PG(3, q), and any line class of such a tactical decomposition is a Cameron-Liebler line class.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

A Characterization

Definition Let L be a set of lines of PG(3, q) with |L| = x(q2 + q + 1), x a nonnegative integer. We say that L is a Cameron-Liebler line class with parameter x if every spread of PG(3, q) contains x lines of L.

1 The complement of L in the set of all lines of PG(3, q) is a

Cameron-Liebler line class with parameter q2 + 1 − x. WLOG we may assume that x ≤ q2+1

2 .

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

“Trivial” examples

Let (P, π) be any non-incident point-plane pair of PG(3, q).

1 star(P): the set of all lines through P, 2 line(π): the set of all lines contained in the plane π.

Example The following are examples of Cameron-Liebler line classes:

1 x = 0: ∅; 2 x = 1: star(P), line(π); 3 x = 2: star(P) ∪ line(π).

It was conjectured by Cameron and Liebler that up to taking complement these are all the examples of Cameron-Liebler line classes.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

More Examples

1 The first counterexample was given by Drudge 3 in PG(3, 3),

and it has parameter x = 5.

2 Bruen and Drudge (1999)4 generalized the above example

into an infinite family with parameter x = q2+1

2

for all odd q.

3 Govaerts and Penttila (2005)5 gave a sporadic example with

parameter x = 7 in PG(3, 4).

• 3K. Drudge, On a conjecture of Cameron and Liebler, Europ. J. Combin.,20 (1999), 263–269.

4A.A. Bruen, K. Drudge, The construction of Cameron-Liebler line classes in PG(3, q), Finite Fields Appl., 5 (1999), 35–45.

• 5P. Govaerts, T. Penttila, Cameron-Liebler line classes in PG(3, 4), Bull. Belg. Math. Soc. Simon Stevin, 12

(2005), 793–804.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

I am going to talk about...

1 We construct a new infinite family of Cameron-Liebler line

classes with parameter x = q2−1

2

for all q ≡ 5 or 9 (mod 12).

2 In the case where q is an even power of 3, we construct the

first infinite family of affine two-intersection sets, whose existence was conjectured by Rodgers. I should remark that De Beule, Demeyer, Metsch and Rodgers also

• btained the same results independently at about the same time by

using a more geometric approach.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

Nonexistence

1 Penttila (1991): x = 3 for all q, and x = 4 for q ≥ 5. 2 Drudge (1999): 2 < x < √q. 3 Govaerts and Storme (2004): 2 < x ≤ q, q prime. 4 De Beule, Hallez and Storme (2008): 2 < x ≤ q/2. 5 Metsch (2010): 2 < x ≤ q 6 Metsch (2014): 2 < x < q 3

• q

2 − 2 3q.

7 Gavrilyuk and Metsch (2014): A modular equality for

Cameron-Liebler line classes It seems reasonable to believe that for any fixed 0 < ǫ < 1 and constant c > 0 there are no Cameron-Liebler line classes with 2 < x < cq2−ǫ for sufficiently large q.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

The Klein correspondence

Let ℓ = u, v be a line of PG(3, q), where u = (ui)0≤i≤3 and v = (vi)0≤i≤3. We define a point θ(ℓ) of PG(5, q) as follows: θ(ℓ) = l01, l02, l03, l12, l13, l23, lij = uivj − ujvi. Note that θ(ℓ) is independent of the choice of the basis u, v. Definition The Klein correspondence is the above map θ : {lines of PG(3, q)} → PG(5, q). Its image set is the Klein quadric Q+(5, q) := {l01, l02, l03, l12, l13, l23 : l01l23 − l02l13 + l03l12 = 0}.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

x-tight sets of Q+(5, q)

Definition A subset M of Q+(5, q) is called an x-tight set if for every point P ∈ Q+(5, q), |P ⊥ ∩ M| = x(q + 1) + q2 or x(q + 1) according as P is in M or not, where ⊥ is the polarity determined by Q+(5, q).

1 Important observation: It holds that |P ⊥ ∩ M| = x(q + 1) for

any point P off Q+(5, q). Consequently M is a projective two-intersection set in PG(5, q) with intersection sizes h1 = x(q + 1) + q2 and h2 = x(q + 1).

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

Cameron-Liebler line classes, another characterization

A line set L of PG(3, q) is a Cameron-Liebler line class with parameter x iff its image M under the Klein correspondence is an x-tight set in the Klein quadric Q+(5, q). Definition Let L be a set of x(q2 + q + 1) lines in PG(3, q), with 0 < x ≤ q2+1

2 , and M be the image of L under the Klein

• correspondence. Then L is a Cameron-Liebler line class with

parameter x iff it holds that |P ⊥ ∩ M| =

• x(q + 1) + q2,

if P ∈ M, x(q + 1),

• therwise.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

character values vs. hyperplane intersections

Let M be a subset of PG(5, q). We define D := {λv : λ ∈ F∗

q, v ∈ M} ⊂ (F6 q, +).

Let ψ be a nonprincipal additive character of F6

• q. Then ψ is

principal on a unique hyperplane P ⊥ for some P ∈ PG(5, q). ψ(D) =

• v∈M
• λ∈F∗

q

ψ(λv) =

• v∈M

(q[[v ∈ P ⊥]] − 1) = −|M| + q|P ⊥ ∩ M|. Therefore, the character values of D reflect the intersection properties of M with the hyperplanes of PG(5, q).

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

Cameron-Liebler line classes, yet another characterization

Definition Let L be a set of x(q2 + q + 1) lines in PG(3, q) and M be the image of L under the Klein correspondence. Define D := {λv : λ ∈ F∗

q, v ∈ M} ⊂ (F6 q, +).

Then L is a Cameron-Liebler line class with parameter x iff |D| = (q3 − 1)x and for any P ∈ PG(5, q) ψ(D) =

• −x + q3,

if P ∈ M, −x,

• therwise,

where ψ is any nonprincipal character of F6

q that is principal on P ⊥.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

Work by Rodgers

Recent work by Rodgers suggests that there are probably more infinite families awaiting to be discovered.

1 x = q2−1

2

for q ≡ 5 or 9 (mod 12) and q < 200;6

2 x = (q+1)2

3

for q ≡ 2 (mod 3) and q < 150.7

3 The first step in our construction follows the same idea as in

Rodgers’ thesis.

• 6M. Rodgers, Cameron-Liebler line classes, Des. Codes Crypto., 68 (2013), 33–37.
• 7M. Rodgers, On some new examples of Cameron-Liebler line classes, PhD thesis, University of Colorado,

2012.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

A model of Q+(5, q)

Let E = Fq3 and F = Fq. We view E × E as a 6-dimensional vector space over F. Define a quadratic form Q : E × E → F by Q((x, y)) = Tr(xy), ∀ (x, y) ∈ E × E. The quadratic form Q is nondegenerate and {(x, 0) | x ∈ E} is a totally isotropic subspace, and so the quadric defined by Q is

• hyperbolic. This is our model of Q+(5, q).

Remark For a point P = (x0, y0), its polar hyperplane P ⊥ is given by P ⊥ = {(x, y) : Tr(xy0 + x0y) = 0}.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

Prescribing an automorphism group of M

We need to construct a subset M of Q+(5, q) with the desired hyperplane intersection properties. Let ω be a primitive element of E = Fq3, and ω1 = ωq−1. Assuming that q ≡ 5 or 9 (mod 12). Definition Define the map g on Q+(5, q) by g : (x, y) → (ω1x, ω−1

1 y),

where ω1 ∈ E∗ has order N = q2 + q + 1. Then the cyclic subgroup C ≤ PGO+(6, q) generated by g acts semi-regularly on the points of Q+(5, q); each orbit of C has length q2 + q + 1. The x-tight set M we intend to construct will be a union of x orbits of C acting on Q+(5, q).

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

The construction

Definition With X a proper subset of Z2N of size q + 1, we define IX : = {2i : i ∈ X} ∪ {2i + N : i ∈ X} ⊂ Z4N, D : = {(xy, xy−1zωℓ) | x ∈ F ∗, y ∈ ωq−1, z ∈ ω4N, ℓ ∈ IX}. It is our purpose to find the correct X ⊂ Z2N such that ψa,b(D) =

• − q2−1

2

+ q3, if (b, a) ∈ D, − q2−1

2 ,

• therwise,

for all (0, 0) = (a, b) ∈ E × E.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

The exponential sums

Let S be the set of nonzero squares of F. The computation of the character values of D is essentially reduced (by using complicated computations involving Gauss sums) to the computation of Tu :=

• i∈X

ψF

• Tr(ωu+i)S
• , 0 ≤ u ≤ q3 − 2.

The next step is to describe X and explicitly determine the Tu’s. This is accomplished by some geometric arguments.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

The set X

Consider the conic Q = {x : Tr(x2) = 0} in the plane PG(2, q), and define IQ := {i : 0 ≤ i ≤ N − 1, Tr(w2i

1 ) = 0} = {d0, d1, . . . , dq}.

where the elements are numbered in any (unspecified) order. Definition For d0 ∈ IQ, we define X := {wdi

1 Tr(wd0+di 1

) : 1 ≤ i ≤ q} ∪ {2wd0

1 }

and X := {logω(x) (mod 2N) : x ∈ X} ⊂ Z2N, .

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

The following lemma indicates that the choice of d0 is irrelevant. Lemma

1 Let di, dj, dk be three distinct elements of IQ. Then

2Tr(wdi+dj

1

)Tr(wdi+dk

1

)Tr(wdj+dk

1

) is a nonzero square of Fq.

2 If we use any other di in place of d0 in the definition of X,

then the resulting set X′ satisfies that X′ ≡ X (mod 2N) or X′ ≡ X + N (mod 2N), and correspondingly the value of Tu is either unchanged or is equal to Tu+N. The proof makes use of the fact that the determinants of the Gram matrices of the associated bilinear form w.r.t. two distinct basis differ by a nonzero square.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

The main result

With the above choice of X, the exponential sums Tu’s are explicitly determined by some geometric arguments. We thus have the following main result. Theorem Let M be the set of projective points in PG(5, q) corresponding to

• D. Then |M| = q2−1

2 (q2 + q + 1) and M ⊂ Q+(5, q). The line set

L in PG(3, q) corresponding to M under the Klein correspondence forms a Cameron-Liebler line class with parameter x = q2−1

2 .

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

m-Ovoids of Q(4, q)

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

m-Ovoids of Q(4, q)

Definition A set O of points of Q(4, q) is called an m-ovoid if every line of Q(4, q) meets O in m points. For odd q, there exist known m-ovoids of Q(4, q) for three values

• f m:

1 m = 1. 1-ovoids are usually called ovoids. For example,

Q−(3, q) is an ovoid of Q(4, q).

2 m = q. The complement of an ovoid of Q(4, q) is a q-ovoid. 3 m = q+1

2 . Hyperplane section of a hemisystem of Q−(5, q).

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

New m-ovoids with m = q−1

2

Very recently, Tao Feng, Koji Momihara and Qing Xiang constructed a family of q−1

2 -ovoid of Q(4, q) when q ≡ 3 (mod 4).

The idea is very similar to the one we used in the construction of Cameron-Liebler line classes: We prescribe an automorphism group for the m-ovoids that we intend to construct, and then take union

• f orbits of the point set of Q(4, q) under the action of the

prescribed automorphism group.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

New ingredient

The new ingredient in our approach to the construction of m-ovoids is that we prescribe an automorphism group of medium

• size. Consequently the number of orbits of the action of the group

is large, and geometric argument for analyzing the intersection of the orbits with hyperplanes seems impossible. We have to come up algebraic description of the m-ovoids, and use algebraic techniques to prove the intersection property with lines.

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

Further Work

1 Generalize the examples with parameter x = (q+1)2

3

for q ≡ 2 (mod 3) and q < 150 into an infinite family;

2 Find an infinite family of Cameron-Liebler line classes in the

even characteristic;

3 Obtain more nonexistence results on Cameron-Liebler line

classes;

4 Construct q−1

2 -ovoids of Q(4, q) when q ≡ 1 (mod 4).

Outline General Remarks Finite Classical Polar Spaces m-ovoids and x-tight sets Cameron-Liebler line classes m-Ovoids of Q(4

Thank You !