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 1 T. Feng, K. Momihara, Q. Xiang, Cameron-Liebler line classes with parameters x = q 2 − 1 , J. Combin. Theory (A), 133 (2015), 307–338 2

Outline General Remarks Finite Classical Polar Spaces m -ovoids and x -tight sets Cameron-Liebler line classes m -Ovoids of Q (4 General Remarks 1 Finite Classical Polar Spaces 2 m -ovoids and x -tight sets 3 Cameron-Liebler line classes 4 m -Ovoids of Q (4 , q ) 5

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 other 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 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 .

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, t 2 ) | t ∈ F q } ∪ { (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 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 .

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 x 2 Q (2 n, q ) n 0 + x 1 x 2 + · · · + x 2 n − 1 x 2 n Q + (2 n + 1 , q ) n + 1 x 0 x 1 + x 2 x 3 + · · · + x 2 n x 2 n +1 Q − (2 n + 1 , q ) n f ( x 0 , x 1 ) + x 2 x 3 + · · · + x 2 n x 2 n +1 W (2 n + 1 , q ) n + 1 x 0 y 1 + y 0 x 1 + · · · + x 2 n y 2 n +1 + x 2 n +1 y 2 n x q +1 + · · · + x q +1 H (2 n, q 2 ) n 0 2 n x q +1 + · · · + x q +1 H (2 n + 1 , q 2 ) n + 1 0 2 n +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 F q . Denote by θ n ( q ) := q n − 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 � xθ r − 1 ( q ) + q r − 1 , if P ∈ M , | P ⊥ ∩ M| = xθ r − 1 ( q ) , otherwise .

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 q 2 + 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 |M| = q r − 1 � q − 1 = θ r − 1 ( q ) + q r − 1 , if P ∈ M , | P ⊥ ∩ M| = θ r − 1 ( q ) , otherwise .

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 ( 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 2 + 1 − x . WLOG we may assume that x ≤ q 2 +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 = q 2 +1 for all odd q . 2 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 = q 2 − 1 for all q ≡ 5 or 9 (mod 12) . 2 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.