Cécile Huybrechts

Dimensional dual hyperovals in projective spaces and c.AG*-geometries

The two subjects of the title are studied, first independently and then by making them interact. Many questions arise from this interaction.In our intrinsic study of the first subject, we construct a new family of examples and define the notions of quotient and universality. For the second subject, we restrict the class c.AG* of circular extensions of dual affine spaces under the two geometrical conditions (LL) and (T) by showing that, apart from some extreme cases, every such rank three geometry can be erected into a rank 4 geometry. In particular, c.AG*n(q)-geometries satisfying (LL) and (T) with n ≥ 3 must have q even. We then deduce similar results for the first subject.

A (c. L*-Geometry for the Sporadic Group J2

We prove the existence of a rank three geometry admitting the Hall–Janko group J2 as flag-transitive automorphism group and Aut(J2) as full automorphism group. This geometry belongs to the diagram (c·L*) and its nontrivial residues are complete graphs of size 10 and dual Hermitian unitals of order 3.

The non-existence of 3-dimensional locally projective spaces of orders (2,9)

Abstract We use results on maximal arcs and their relationship with locally projective spaces to deduce the non-existence of a 3-dimensional locally projective space of orders (2, 9).

On c^{n-2}.c Geometries of Order 2

where q is a positive integer, the label c denotes the class of circular spaces (that is, the class of complete graphs) and c∗ is its dual. A cn−2.c∗ geometry (of order q) is a geometry belonging to the above diagram. (In particular, a c0.c∗ geometry is the dual of a circular space.) The symbol c1.c∗ is usually abbreviated as c.c∗. The diagram c.c∗ covers biplanes and semibiplanes [7]. Flag-transitive geometries belonging to it with arbitrary order q have been studied by Baumeister ([1], [2], [3]); also by Grams and Meixner [6] when q = 4. A classification of flag-transitive c2.c∗ geometries with any order q has been achieved by Meixner [11]. We do not assume flag-transitivity in this paper, but we consider a weaker property, which we call homogeneity: we say that a geometry is homogeneous if all its residues of the same type are mutually isomorphic. We classify homogeneous cn−2.c∗ geometries of order q = 2 (§1.2, Theorem 1.1). As a by-product of that classification it will turn out that the homogeneous cn−2.c∗ geometries of order 2 are precisely the flag–transitive ones.

A Characterization of the Hall-Janko Group J 2 by a c. L *-Geometry

A c.U*-geometry is a geometry over the diagram c.L*, the point residues of which are finite dual unitals. Only one flag-transitive example is known. Its full automorphism group is Aut(J 2), but J2 also acts flag-transitively on it. We shall prove that this geometry is indeed the unique flag-transitive c.U*-geometry, thus obtaining a new geometric characterization of the Hall-Janko group J 2.

An Arithmetic Reduction of Finite Rank 3 Geometries with Linear Spaces as Plane Residues and with Dual Linear Spaces as Point Residues

Let Γ be a rank three incidence geometry of points, lines and planes whose planes are linear spaces and whose point residues are dual linear spaces (notice that we do not require anything on the line residues). We assume that the residual linear spaces of Γ belong to a natural class of finite linear spaces, namely those linear spaces whose full automorphism group acts flag-transitively and whose orders are polynomial functions of some prime number. This class consists of six families of linear spaces. In Γ the amalgamation of two such linear spaces imposes an equality on their orders leading, in particular, to a series of diophantine equations, the solutions of which provide a reduction theorem on the possible amalgams of linear spaces that can occur in Γ.We prove that one of the following holds (up to a permutation of the words “point” and “plane”):A) the planes of Γ and the dual of the point residues belong to the same family and have the same orders,B) the diagram of Γ is in one of six families,C) the diagram of Γ belongs to a list of seven sporadic cases.Finally, we consider the particular case where the line residues of Γ are generalized digons.

L L -geometries and Dn-buildings

Abstract A locally finite PG · PG∗ - geometry is a rank three incidence structure of points, lines and blocks , the block and dual point residues of which are {point,line}-systems of finite projective geometries and the line residues of which are generalized digons. We show that every locally finite thick PG · PG*-geometry satisfying two geometrical axioms (LL) and (T) is a truncation of a D n -building. We also strongly restrict geometries satisfying (LL) and (T) over the more general diagram L · PG*.