isibang/ms/2013/2 January 1st, 2013 http://www.isibang.ac.in/e statmath/eprints Ovoidal packings of PG(3; q) for even q Bhaskar Bagchi and N.S. Narasimha Sastry Indian Statistical Institute, Bangalore Centre 8th Mile Mysore Road, Bangalore, 560059 India Ovoidal packings of PG(3; q) for even q Bhaskar Bagchi∗ and N.S. Narasimha Sastryy Theoretical Statistics and Mathematics Unit Indian Statistical Institute Bangalore - 560 059 India. Abstract We show that any set of n pairwise disjoint ovals in a finite pro- jective plane of even order has a unique common tangent. As a con- sequence, any set of q + 1 pairwise disjoint ovoids in PG(3; q), q even, has exactly q2 +1 common tangent lines, constituting a regular spread. Also, if q −1 ovoids in PG(3; q) intersect pairwise exactly in two given points x =6 y and share two tangent planes πx; πy at these two points, then these ovoids share exactly (q + 1)2 common tangent lines, and they consist of the transversals to the pair xy, πx \ πy of skew lines. There is a similar (but more complicated) result for the common tan- gent lines to q ovoids in PG(3; q) which are mutually tangent at a common point and share a common tangent plane through this point. It is also shown that the common tangent lines to any pair of disjoint ovoids of PG(3; q), q even, form a regular spread. 1 Introduction Let us recall that an oval in a finite projective plane of order n is a set of n + 1 points no three of which are collinear. It follows that an ∗email id: [email protected] [email protected] 1 oval has exactly n + 1 tangent lines, one such through each point of the oval. When A is an oval in a projective plane π of odd order n, then the set A∗ of the tangent lines to A form an oval in the dual projective plane π∗; we have A∗∗ = A. However, when n is even, any oval A in a plane of order n has a unique point x 62 A such that the tangent lines to A are precisely the lines through x. The point x is called the nucleus of A. When q is an odd prime power, a famous theorem due to Segre says that the ovals in the Desarguesian plane PG(2; q) are precisely the conics (i.e., the set of all points, the homogeneous co-ordinates of which satisfy homogeneous quadratic equation). However, when q = 2e is large, classification of ovals in PG(2; q) is a hopelessly difficult problem. For prime powers q, an ovoid in the three-dimensional projective space PG(3; q) is a set of points which meet every plane in a point or an oval. A trivial count shows that there are exactly q2 + 1 points on it, and it has exactly q2 + 1 tangent planes, one at each of its point. Indeed, the set O∗ of tangent planes to O form an ovoid in the dual of PG(3; q). When q > 2, the ovoids in PG(3; q) are precisely the point sets of PG(3; q) of the largest possible size no three of which are collinear. When q is an odd prime power, there is a beautiful three-dimensional analogue, due to Barlotti and Panella (see [F], 1.4.50) of Segre's the- orem. In this case, the ovoids in PG(3; q) are precisely the elliptic quadrics (i.e., non-degenerate quadrics of Witt index one). The situation with ovoids is more interesting for q = 2e. When e = 1 or e is even, the elliptic quadrics are the only known ovoids in PG(3; q). When e > 1 is odd, apart from the elliptic quadrics, there is only one more isomorphism class of known ovoids, namely the Tits ovoids. Despite serious attempts by many researchers, no other examples of ovoids have turned up. Thus we have the following conjecture, which is perhaps the most tantalising open problem in finite geometry. Conjecture 1.1 Up to isomorphism, the elliptic quadrics and Tits ovoids are the only ovoids in PG(3; q). This conjecture gains extra importance due to a related result of Dembowski. Recall that a finite inversive plane of order n > 1 is nothing but a Steiner 3-design with parameters (n2 + 1; n + 1; 1). In other words, it is an one-point extension of an affine plane of order n. 2 A well-known but apparently inaccessible conjecture asserts that the order of any finite affine plane (equivalently, a finite projective plane) is a prime power. The corresponding problem for inversive planes may be more accessible. Indeed, the only known finite inversive planes are the \egg-like" ones, i.e., those which arise as the non-trivial plane sections of ovoids in PG(3; q). We still do not know if all finite inversive planes are egg-like. But this is a theorem of Demboswki in the even order case. If the order of a finite inversive plane is even, then indeed it is a power of 2, and the points and blocks of the inversive plane are (up to isomorphism) the points and non-trivial plane sections of an ovoid in PG(3; q). Thus, the classification of inversive planes of even order is equivalent to the classification of ovoids in PG(3; q), q even. It is our hope that if we could gain a good understanding of how the ovoids (known or unknown) must intersect, then it would be possible to see that there is no \space" in PG(3; q) for any unknown ovoid. This motivates our previous papers [1], [2] as well as the current paper. Recall that O∗ is the dual ovoid consisting of the tangent planes to a given ovoid O. The following remarkable result is due to Bruen and Hirschfield [4]. We cannot resist the temptation to reproduce its simple and elegant proof. Let O1 and O2 be two ovoids in PG(3; q). 2 O 2 O∗ Let us count the number of pairs (x; π) where x 1; π 2 and x 2 π. Since each point on O2 is on a unique tangent plane to O2 and each point off O2 is on q + 1 tangent planes to O2, the number of 2 such pairs is #(O1 \O2) + (q + 1)(q + 1 − #(O1 \O2)). A dual way O∗ $ O O∗ $ O of counting these pairs (effect the interchanges 2 1, 1 2) O∗ \O∗ 2 − O∗ \O∗ yields #( 1 2) + (q + 1)(q + 1 #( 1 2)) as their number. Equating the two answers, we get: Theorem 1.2 (Bruen and Hirschfield): For any two ovoids O1 and O O∗ \O∗ O \O 2 in PG(3; q), we have #( 1 2) = #( 1 2). Notice that this result is valid for any prime power q (even or odd). This result is all the more remarkable since its analogue for ovals is false. (For instance, there are disjoint ovals sharing a tangent, compare Theorem 2.1 below.) So this result cries out for a geometric explanation. In this context, it seems very natural to ask. Question 1.3 Is it true that any common point of two ovoids in PG(3; q) lie on a common tangent plane to them? 3 Notice that, if the answer to this questions was in the affirmative, then the map x 7! πx, sending any common point x to the unique com- mon tangent plane πx through x, would provide a bijection between O \O O∗ \O∗ 1 2 and 1 2, thus explaining Theorem 1.2. Unfortunately, we find that the answer to this question is in the negative, at least for q > 2, even. To explain our answer, let us introduce the following notation. For ovoids O1; O2, let I(O1; O2) be the set of all pairs (x; π) 2 O \O 2 O∗ \O∗ 2 where x 1 2, π 1 2 and x π. In Section 3 of this e paper, we show that for q = 2 and ovoids O1; O2 of PG(3; q) which do not share all tangent lines, we have #I(O1; O2) ≤ 2. Therefore, if Question 1.3 had an affirmative answer, then it would follow that we have #(O1 \O2) ≤ 2 for any such pair of ovoids. But this is easily disproved when q > 2. So we are forced to ask the following vague question. Revised Question 1.4 For ovoids O1; O2 in PG(3; q), is there a ge- O \O O∗\O∗ ometrically defined (\natural") bijection between 1 2 and 1 2? We recall that a linear complex in PG(3; q) is the set of all ab- solute lines with respect to a symplectic polarity of PG(3; q) (i.e. a polarity for which all points are absolute). If P is the set of all points of PG(3; q) and L is a linear complex in PG(3; q) then the incidence system (P; L) is denoted by W (q). Since the action of the collineation group P GL(4; q) by conjugation is transitive on the set of all sym- plectic polarities, the incidence system W (q) is uniquely defined, up to isomorphism. It is a generalized quadrangle of order (q; q) (Cf. [Payne + Thas]). (An...) ovoid of W (q) is a set of q2 + 1 points no two of which are collinear. It is easy to see that every ovoid of W (q) is an ovoid of PG(3; q).