Tight Sets and m-ovoids of Quadrics 1
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1 Tight Sets and m-ovoids of Quadrics 1 Qing Xiang Department of Mathematical Sciences University of Delaware Newark, DE USA xiang@mathudeledu Joint work with Tao Feng and Koji Momihara 1 T Feng, K Momihara, Q Xiang, Cameron-Liebler line classes with parameters x = q2 1, J Combin Theory (A), 133 (2015),
2 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)
3 General Remarks
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 other combinatorial objects, such as translation planes, designs, strongly regular graphs, two-weight codes, and association schemes, etc
5 A sample result Let V be an (n + 1)-dimensional vector space over F q We will consider PG(n, q), whose points are the 1-dimensional subspaces of 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 of 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
6 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, t 2 ) t F q } {(0, 0, 1)} In contrast, ovals (and hyperovals) in PG(2, q), q even, are far from being classified
7 Finite Classical Polar Spaces
8 Definitions Let V (n + 1, q) be an (n + 1)-dimensional vector space over F q, 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
9 A polar space S contains the totally isotropic points, lines, planes, etc of the ambient projective space The generators of S are the (ti) 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
10 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 x x 1x x 2n 1 x 2n Q + (2n + 1, q) n + 1 x 0 x 1 + x 2 x x 2n x 2n+1 Q (2n + 1, q) n f(x 0, x 1 ) + x 2 x x 2n x 2n+1 W (2n + 1, q) n + 1 x 0 y 1 + y 0 x x 2n y 2n+1 + x 2n+1 y 2n H(2n, q 2 ) n x q x q+1 2n H(2n + 1, q 2 ) n + 1 x q x q+1 2n+1
11 m-ovoids and x-tight sets
12 Let S be a finite classical polar space of rank r over the finite field F q Denote by θ n (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 xθ r 1 (q) + q r 1, if P M, M = xθ r 1 (q), otherwise
13 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 q 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 = q r 1 q 1 = θ r 1(q) + q r 1, if P M, θ r 1 (q), otherwise
14 Cameron-Liebler Line Classes
15 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 2 PJ Cameron, RA Liebler, Tactical decompositions and orbits of projective groups, Linear Algebra Appl, 46 (1982),
16 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
17 A Characterization Definition Let L be a set of lines of PG(3, q) with L = x(q 2 + 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 q x WLOG we may assume that x q2 +1 2
18 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
19 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 = q for all odd q 3 Govaerts and Penttila (2005) 5 gave a sporadic example with parameter x = 7 in PG(3, 4) 3 K Drudge, On a conjecture of Cameron and Liebler, Europ J Combin,20 (1999), AA Bruen, K Drudge, The construction of Cameron-Liebler line classes in PG(3, q), Finite Fields Appl, 5 (1999), P Govaerts, T Penttila, Cameron-Liebler line classes in PG(3, 4), Bull Belg Math Soc Simon Stevin, 12 (2005),
20 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 obtained the same results independently at about the same time by using a more geometric approach
21 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 q 7 Gavrilyuk and Metsch (2014): A modular equality for Cameron-Liebler line classes It seems reasonable to believe that for any fixed 0 < ϵ < 1 and constant c > 0 there are no Cameron-Liebler line classes with 2 < x < cq 2 ϵ for sufficiently large q
22 The Klein correspondence Let l = u, v be a line of PG(3, q), where u = (u i ) 0 i 3 and v = (v i ) 0 i 3 We define a point θ(l) of PG(5, q) as follows: θ(l) = l 01, l 02, l 03, l 12, l 13, l 23, l ij = u i v j u j v i 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) := { l 01, l 02, l 03, l 12, l 13, l 23 : l 01 l 23 l 02 l 13 + l 03 l 12 = 0}
23 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) + q 2 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 h 1 = x(q + 1) + q 2 and h 2 = x(q + 1)
24 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(q 2 + 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 x(q + 1) + q 2, if P M, M = x(q + 1), otherwise
25 character values vs hyperplane intersections Let M be a subset of PG(5, q) We define D := {λv : λ F q, v M} (F 6 q, +) Let ψ be a nonprincipal additive character of F 6 q Then ψ is principal on a unique hyperplane P for some P PG(5, q) ψ(d) = ψ(λv) = (q[[ v P ]] 1) v M λ F q = M + q P M v M Therefore, the character values of D reflect the intersection properties of M with the hyperplanes of PG(5, q)
26 Cameron-Liebler line classes, yet another characterization Definition Let L be a set of x(q 2 + 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} (F 6 q, +) Then L is a Cameron-Liebler line class with parameter x iff D = (q 3 1)x and for any P PG(5, q) { x + q 3, if P M, ψ(d) = x, otherwise, where ψ is any nonprincipal character of F 6 q that is principal on P
27 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 < The first step in our construction follows the same idea as in Rodgers thesis 6 M Rodgers, Cameron-Liebler line classes, Des Codes Crypto, 68 (2013), M Rodgers, On some new examples of Cameron-Liebler line classes, PhD thesis, University of Colorado, 2012
28 A model of Q + (5, q) Let E = F q 3 and F = F q 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 = (x 0, y 0 ), its polar hyperplane P is given by P = { (x, y) : Tr(xy 0 + x 0 y) = 0}
29 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 = F q 3, and ω 1 = ω q 1 Assuming that q 5 or 9 (mod 12) Definition Define the map g on Q + (5, q) by g : (x, y) (ω 1 x, ω 1 1 y), where ω 1 E has order N = q 2 + q + 1 Then the cyclic subgroup C P GO + (6, q) generated by g acts semi-regularly on the points of Q + (5, q); each orbit of C has length q 2 + 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)
30 The construction Definition With X a proper subset of Z 2N of size q + 1, we define I X : = {2i : i X} {2i + N : i X} Z 4N, D : = {(xy, xy 1 zω l ) x F, y ω q 1, z ω 4N, l I X } It is our purpose to find the correct X Z 2N such that { q2 1 ψ a,b (D) = 2 + q 3, if (b, a) D, q2 1 2, otherwise, for all (0, 0) (a, b) E E
31 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 T u := i X ψ F ( Tr(ω u+i )S ), 0 u q 3 2 The next step is to describe X and explicitly determine the T u s This is accomplished by some geometric arguments
32 The set X Consider the conic Q = { x : Tr(x 2 ) = 0} in the plane PG(2, q), and define I Q := {i : 0 i N 1, Tr(w 2i 1 ) = 0} = {d 0, d 1,, d q } where the elements are numbered in any (unspecified) order Definition For d 0 I Q, we define X := {w d i 1 Tr(wd 0+d i 1 ) : 1 i q} {2w d 0 1 } and X := {log ω (x) (mod 2N) : x X} Z 2N,
33 The following lemma indicates that the choice of d 0 is irrelevant Lemma 1 Let d i, d j, d k be three distinct elements of I Q Then 2Tr(w d i+d j 1 )Tr(w d i+d k 1 )Tr(w d j+d k 1 ) is a nonzero square of F q 2 If we use any other d i in place of d 0 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 T u is either unchanged or is equal to T u+n The proof makes use of the fact that the determinants of the Gram matrices of the associated bilinear form wrt two distinct basis differ by a nonzero square
34 The main result With the above choice of X, the exponential sums T u 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 (q 2 + 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
35 m-ovoids of Q(4, q)
36 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 of 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)
37 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 of orbits of the point set of Q(4, q) under the action of the prescribed automorphism group
38 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
39 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 -ovoids of Q(4, q) when q 1 (mod 4) 2
40 Thank You!
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