Joseph A. Thas

General Galois geometries

This book is the second edition of the third and last volume of a treatise on projective spaces over a finite field, also known as Galois geometries. This volume completes the trilogy comprised of plane case (first volume) and three dimensions (second volume). This revised edition includes much updating and new material. It is a mostly self-contained study of classical varieties over a finite field, related incidence structures and particular point sets in finite n-dimensional projective spaces. General Galois Geometries is suitable for PhD students and researchers in combinatorics and geometry. The separate chapters can be used for courses at postgraduate level.

Generalized quadrangles and flocks of cones

A flock of the quadratic cone K of PG(3, q) is a partition of K but its vertex into disjoint conics. If the planes of the q conics of such a flock all contain a common line, then the flock is called linear. With any flock there corresponds a translation plane which is Desarguesian iff the flock is linear. W. M. Kantor showed that with a set of q upper triangular 2 × 2-matrices over GF(q) of a certain type, there corresponds a generalized quadrangle of order (q2, q). We prove that with such a set of q matrices there corresponds a flock of the quadratic cone of PG(3, q), and conversely that with each flock of the quadratic cone there corresponds such a set of matrices. Using this relationship, new flocks, new generalized quadrangles, and probably new translation planes are obtained.

Ovoids and spreads of finite classical polar spaces

LetP be a finite classical polar space of rankr, r⩾2. An ovoidO ofP is a pointset ofP, which has exactly one point in common with every totally isotropic subspace of rankr. It is proved that the polar spaceWn(q) arising from a symplectic polarity ofPG(n, q), n odd andn > 3, that the polar spaceQ(2n, q) arising from a non-singular quadric inPG(2n, q), n > 2 andq even, that the polar space Q−(2n + 1,q) arising from a non-singular elliptic quadric inPG(2n + 1,q), n > 1, and that the polar spaceH(n,q2) arising from a non-singular Hermitian variety inPG(n, q2)n even andn > 2, have no ovoids.LetS be a generalized hexagon of ordern (⩾1). IfV is a pointset of order n3 + 1 ofS, such that every two points are at distance 6, thenV is called an ovoid ofS. IfH(q) is the classical generalized hexagon arising fromG2(q), then it is proved thatH(q) has an ovoid iffQ(6, q) has an ovoid. There follows thatQ(6, q), q=32h+1, has an ovoid, and thatH(q), q even, has no ovoid.A regular system of orderm onH(3,q2) is a subsetK of the lineset ofH(3,q2), such that through every point ofH(3,q2) there arem (> 0) lines ofK. B. Segre shows that, ifK exists, thenm=q + 1 or (q + l)/2.If m=(q + l)/2,K is called a hemisystem. The last part of the paper gives a very short proof of Segres result. Finally it is shown how to construct the 4-(11, 5, 1) design out of the hemisystem with 56 lines (q=3).

Spreads and ovoids in finite generalized quadrangles

This paper is a survey on the existence and non-existence of ovoids and spreads in the known finite generalized quadrangles. It also contains the following new results. We prove that translation generalized quadrangles of order (s,s2), satisfying certain properties, have a spread. This applies to three known infinite classes of translation generalized quadrangles. Further a new class of ovoids in the classical generalized quadranglesQ(4, 3e),e≥3, is constructed. Then, by the duality betweenQ(4, 3e) and the classical generalized quadrangleW (3e), we get line spreads of PG(3, 3e) and hence translation planes of order 32e. These planes appear to be new. Note also that only a few classes of ovoids ofQ(4,q) are known. Next we prove that each generalized quadrangle of order (q2,q) arising from a flock of a quadratic cone has an ovoid. Finally, we give the following characterization of the classical generalized quadranglesQ(5,q): IfS is a generalized quadrangle of order (q,q2),q even, having a subquadrangleS′ isomorphic toQ(4,q) and if inS′ each ovoid consisting of all points collinear with a given pointx ofS\S′ is an elliptic quadric, thenS is isomorphic toQ(5,q).

Projective Geometry over a Finite Field

Publisher Summary This chapter focuses on projective geometry over a finite field. A k-arc in projective plane, PG (n, q) is a set K of k points with k ≥ n + 1 such that no n + 1 points of K lie in a hyperplane. An arc K is complete if it is not properly contained in a larger arc. A normal rational curve of PG(2, q) is an irreducible conic; a normal rational curve of PG(3, q) is a twisted cubic. It is well known that any (n + 3)-arc of PG(n, q) is contained in a unique normal rational curve of this space. For q > n + 1, the osculating hyperplane of the normal rational curve C at the point x ϵ C is the unique hyperplane through x intersecting C at x with multiplicity n. In PG(n, q), n ≥ 3, a set K of k points no three of which are collinear is a k-cap. A k-cap is complete if it is not contained in a (k + 1)-cap. A line of PG(n, q) is a secant, a tangent, or an external line of a k-cap as it meets K in 2, 1, or 0 points.

Polar spaces, generalized hexagons and perfect codes

w’,(q): the polar space arising from a symplectic polarity of PG(n, q), n odd; Q(2n, 4): the polar space arising from a non-singular quadric Q in PG(2n, 4); Q*(2n + 1, 4): the polar space arising from a non-singular hyperbolic quadric Q+ [S] in PG(2n + 1, q); Q-(2n + 1, q): the polar space arising from a non-singular elliptic quadric Q[S] in PG(2n + 1, q); H(n, q2): the polar space arising from a non-singular hermitian variety 41 IS] in PG(n, 43.

m -systems of polar spaces

Abstract Let P be a finite classical polar space of rank r , with r ⩾ 2. A partial m -system M of P , with 0 ≤ m ≦ r − 1, is any set { π 1 , π 2 , …, π k } of k (≠ 0) totally singular m -spaces of P such that no maximal totally singular space containing π i has a point in common with ( ν 1 ∪ π 2 ∪ ⋯ ∪ π k ) − π i , i = 1, 2, …, k . In each of the respective cases an upper bound δ for | M | is obtained. If | M | = δ, then M is called an m -system of P . For m = 0 the m -systems are the ovoids of P ; for m = r − 1 the m -systems are the spreads of P . Surprisingly δ is independent of m , giving the explanation why an ovoid and a spread of a polar space P have the same size. In the paper many properties of m -systems are proved. We show that with m -systems of three types of polar spaces there correspond strongly regular graphs and two-weight codes. Also, we describe several ways to construct an m ′-system from a given m -system. Finally, examples of m -systems are given.

Complete Arcs in Planes of Square Order

Large arcs in cyclic planes of square order are constructed as orbits of a subgroup of a group whose generator acts as a single cycle. In the Desarguesian plane of even square order, this gives an example of an are achieving the upper bound for complete arcs other than ovals.

Semi-partial geometries and spreads of classical polar spaces

Abstract A new construction method for semi-partial geometries is given and new examples of semi-partial geometries are deduced. Further, several constructions of spreads of the quadratic Q−(5, q) are given.

Generalized Quadrangles of Order (s, s2), II

Let S=(P, B, I) be a generalized quadrangle of order (q, q2), q>1, and assume that S satisfies Property (G) at the flag (x, L). If q is odd then S is the dual of a flock generalized quadrangle. This solves (a stronger version of ) a ten-year-old conjecture. We emphasize that this is a powerful theorem as Property (G) is a simple combinatorial property, while a flock generalized quadrangle is concretely described using finite fields and groups. As in several previous theorems it was assumed that the dual of the generalized quadrangle arises from a flock, this can now be replaced, in the odd case, by having Property (G) at some flag. Finally we describe a pure geometrical construction of a generalized quadrangle arising from a flock; until now there was only the construction by Knarr which only worked in the odd case.