Let F be a cubic surface with 27 lines in PG(3,q). Theorem 30.1 in Manin [2] states that, if q > 34, then there exists a point of F on none of its lines. There is, however, sufficient information in [1] to work out the precise list of cubic surfaces with no such point.

When the cubic curves of PG(2,q) are mapped by their coefficients to the points of PG(9,q), then the set of triple lines is mapped to the Del Vezzo surface v29. Successive projections, always from points of themselves, are also called Del Pezzo surfaces by Manin. A cubic surface with 27 lines in PG(3,q) is in this sense a Del Pezzo surface of order three, whose lines are its exceptional curves. In what follows, F is always such a surface. The 27 lines of F lie by threes in 45 tritangent planes, in e of which the three lines are concurrent at an Eckardt point. All the subsequent lemmas come from [1].

LEMMA 1: F exists over all fields except GF(q) with q = 2, 3 or 5.

LEMMA 2: The number of points on F is q2 + 7q + 1.

LEMMA 3: The number of points on the 27 lines of F is 27(q - 4) + e.

LEMMA 4: For q odd, e ≥ 18; for q even, e ≥ 45.

We consider (s, r; μ)-nets admitting a group G of auto-morphisms acting regularly on the point set and fixing each parallel class; such nets will be called translation nets (due to Sprague for μ = 1). Our main interest is in deriving bounds on the maximum possible value of r (given s and μ) subject to certain restrictions on G (e.g. for abelian groups). Translation nets are equivalent to a generalization of the congruence partitions defined by André. We prove a decomposition theorem in the case of nilpotent groups and show that here the problem may be reduced to finding the maximum value of r for (pi, r; pj)-translation nets with elementary abelian translation group; this is related to partial t-spreads. Using a result of Schulz, we show that every translation affine design has the parameters of an affine space or desarguesian affine plane and has an elementary abelian translation group. A similar result holds for symmetric translation nets with a nilpotent translation group.

INTRODUCTION AND PRELIMINARY KNOWLEDGE

In this paper we consider a generalization of the well-known nets of Bruck [4], [5] where non-parallel blocks intersect μ times (μ not necessarily equal to 1); we are interested in such structures admitting a translation group G (i.e. G acts regularly on the point set and fixes each parallel class).

INTRODUCTION

We have obtained a necessary and sufficient condition for a geometry to be embeddable in a (generalized) projective space. This condition essentially requires that all intervals above a point are embeddable in such a space and that these embeddings are “compatible” along the lines of the geometry.

All geometries considered here are geometric lattices; embeddings are isometric in Kantor's sense (see [14]).

This result provides a unique general proof of the following known theorems on locally embeddable geometries of dimension at least four: Mäurer's result on locally affine geometries (the socalled Möbius spaces), Kantor's strong embedding theorem and his other embedding theorems involving universal properties (see [16] and [13], [14]). The result can be extended to the dimension three case, generalizing Kahn's work on Mobius, Laguerre and Minkowski planes (see [8] to [11]).

Let us mention one of its new applications: Kantor's concept of strong embedding can be extended to a larger class of geometries; for instance, it is possible to prove the embeddability of infinite locally affino-projective geometries of dimension at least four.

The notions of geometry and embedding are defined in §2 and §3 respectively. The main result can be found in §4, where more historical details and an idea of the proof are given. This proof is based on a general embedding lemma stated in §5. Finally, §6 presents some applications; it also gives a precise connection between the main result in §4 and the known theorems mentioned above.

Distance-regular graphs having intersection number c2 = 1 are point graphs of linear incidence systems. This simple observation plays a crucial role in both the existence proof of a regular near octagon “associated with the Hall-Janko group” whose point graph is the unique automorphic graph with intersection array {10, 8, 8, 2; 1, 1, 4, 5} and the non-existence proof of a distance-regular graph with intersection array {12, 8, 6,…; 1, 1, 2,…}. These results imply a partial answer to a problem put forward by Biggs in 1976.

BASIC NOTIONS

Let Γ = (V,E) be a connected graph of diameter d and let Γi(α) for α є V denote the set of vertices at distance i from α. We recall from [2] that Γ is distance-regular if for any i (0≤i≤d) the numbers bi = |Γi+1(α) ∩ Γ1 (β)| and ci = |Γi-1(α) ∩ Γ1 (β)| do not depend on the choice of α, β such that β є Γi(α). Of course, bd = 0 and cl = 1. Write k = | Γ1(α)|. The array {k, b1, b2,…, bd-1; c1, c2,…, cd} is called the intersection array of Γ.

The graph Γ is called distance-transitive if its automorphism group Aut(Γ) is transitive on each of the classes {{α, β} ⊃ V | β ∈Γi(α)} (0≤i≤d), and Γ is called automorphic (cf. [3]) whenever it is distance-transitive, not a complete graph or a line graph, and has an automorphism group which is primitive on V.

This book contains the proceedings of a conference that took place from 15th to 19th June, 1980, at the White House, Isle of Thorns, Chelwood Gate, England, which is a conference centre run by the University of Sussex.

There are 36 articles in all. The Introduction provides a background for non-specialists and places in context the 35 papers submitted. An asterisk following an author's name in some multi-authored papers indicates the presenter of the paper. At the back of the book are a list of talks given for which there is no paper and a list of participants.

We are indebted to the British Council for supporting some of the participants and to the publishers John Wiley, McGraw Hill, Oxford University Press, Pitman and Springer for supporting a book exhibition.

Above all, we are profoundly grateful to Mrs. Jill Foster of the University of Sussex for the excellent typing of the camera-ready copy from which this book was produced.

A finite group (G,·) of order n is said to be sequenceable if its elements can be arranged in a sequence a0, a1, a2,…, an-1 in such a way that the partial products b0 = a0, b1 = a0a1, b2 = a0a1a2,…, bn-1 = a0a1 … an-1 are all distinct and so are again the elements of G. It is immediately evident that for this to be possible, the first element a0 must be equal to the identity element e of G.

Sequenceable groups arise in connection with the construction of so-called complete latin squares. A latin square on n symbols is called row complete if each of the n(n-l) ordered pairs of distinct symbols occurs in adjacent positions (cells) in exactly one row of the latin square. Since there are n-l pairs of adjacent cells in each row of the square, we get an exact match between the ordered pairs and the places in which they may occur. An example is given in Figure 1. There is an analogous definition of column completeness. A latin square which is both row complete and column complete is called complete. The square given in Figure 1 is a complete latin square.

A practical application of row complete latin squares of small size is to the statistical design of sequential experiments in which several treatments are to be administered in succession to a number of different subjects.

INTRODUCTION

Near 2n-gons, as defined by Shult and Yanushka [9], form a class of geometries including the familiar generalised 2n-gons. It also includes the dual polar spaces, for which a system of axioms was given by Cameron [3] and refined by Shult [8].

The concept of a flat embedding of a geometry in a projective space is abstracted from the classification of antiflag transitive collineation groups of projective spaces by Cameron and Kantor [4]. Geometries possessing flat embeddings include symplectic geometries, generalised hexagons of type G2(q), and dual orthogonal geometries of type 0(2n+l, q). The embeddings in the last case can be constructed using spinors. Note that the second and third types are near 2n-gons.

This paper is directed towards determining the flat embeddings of near 2n-gons. Theorems 4.1 and 4.2 give this determination under additional hypotheses. As preliminaries, a summary of the theory of near 2n-gons and the axioms for dual polar spaces is given in Section 2, and the embedding theorem of Cameron and Kantor is stated in Section 3. The final section gives an account of the spinor embedding of dual 0(2n+l, q).

I am grateful to A.L. Wells Jr. for discussions about the topic of this paper.