Antonio Pasini

Extended generalized quadrangles

Extended generalized quadrangles (roughly, connected structures whose every residue is a generalized quadrangle) are studied in some detail, especially those which are uniform or strongly uniform. Much basic structure theory is developed, many examples are given, and something approaching characterization is given for many types.

Locally polar geometries with affine planes

Abstract In this paper we consider partial linear spaces containing a set of subspaces isomorphic to affine planes, such that the lines and these affine planes on a fixed point form a non-degenerate polar spaces of rank at least 2. We obtain a complete classification, provided that the rank is at least 3.

Uniform Hyperplanes of Finite Dual Polar Spaces of Rank 3

Let ? be a finite thick dual polar space of rank 3. We say that a hyperplane H of ? is locally singular (respectively, quadrangular or ovoidal) if H?Q is the perp of a point (resp. a subquadrangle or an ovoid) of Q for every quad Q of ?. If H is locally singular, quadrangular, or ovoidal, then we say that H is uniform. It is known that if H is locally singular, then either H is the set of points at non-maximal distance from a given point of ? or ? is the dual of Q(6, q) and H arises from the generalized hexagon H(q). In this paper we prove that only two examples exist for the locally quadrangular case, arising in Q(6, 2) and H(5, 4), respectively. We fail to rule out the locally ovoidal case, but we obtain some partial results on it, which imply that, in this case, the geometry ?\H induced by ? on the complement of H cannot be flag-transitive. As a bi-product, the hyperplanes H with ?\H flag-transitive are classified.

Flag-transitive extensions ofC n geometries

We consider locally Cn-geometries where all planes are circular spaces (i.e., complete graphs). We call them extensions of Cn-geometries or (c.Cn) geometries, for short. We give a classification of finite flag-transitive (c.Cn) geometries) geometries when n≥3. A classification is given also in the case of n=2 under the hypothesis that residues of points are thick classical generalized quadrangles.

On the simple connectedness of hyperplane complements in dual polar spaces, II

Suppose @D is a dual polar space of rank n and H is a hyperplane of @D. Cardinali, De Bruyn and Pasini have already shown that if n>=4 and the line size is greater than or equal to 4 then the hyperplane complement @D-H is simply connected. This paper is a follow-up, where we investigate the remaining cases. We prove that the hyperplane complements are simply connected in all cases except for three specific types of hyperplane occurring in the smallest case, when the rank and the line size are both 3.

Flag-Transitive Buekenhout Geometries

Publisher Summary This chapter discusses the concept of flag-transitive Buekenhout geometries. More than one half of sporadic simple groups are known to act flag-transitively on finite geometries belonging to Buekenhout diagrams obtained from Coxeter diagrams replacing some of strokes for projective planes with strokes for circular spaces or for dual circular spaces. The chapter recalls that a circular space is a finite linear space with lines of size 2—namely, the system of vertices and edges of a complete graph. Classification theorems exist for some classes of geometries; the chapter surveys some of those theorems, choosing certain diagrams for which a classification is known or at least substantial progresses have been done in that direction. The chapter also discusses some background on Lie diagrams, the Neumaier geometry, spherical and nonspherical diagram, and affine constructions.

On the embeddability of polar spaces

We show that every nondegenerate polar space of rank at least 4 with at least three points on each line can be embedded in a projective space. Together with some results from [9] and [12], this provides a particularly elementary proof that any such polar space is of classical type. Our methods involve the use of geometric hyperplanes and work equally well for spaces of finite or infinite rank.

On locally polar geometries whose planes are affine

We give some contributions to the classification of geometries belonging to the following diagram: (Af. Cn)

The non-existence of ovoids in the dual polar space DW(5, q )

An ovoid of a dual polar space D is a set of points meeting every line of D in exactly one point. In this paper, we consider the dual DW(5, q) of the polar space W(5, q) associated to a non-degenerate alternating form of V(6, q), proving that no ovoids exist in DW(5, q).

Locally singular hyperplanes in thick dual polar spaces of rank 4

We study (i-)locally singular hyperplanes in a thick dual polar space Δ of rank n. If Δ is not of type DQ(2n, K), then we will show that every locally singular hyperplane of Δ is singular. We will describe a new type of hyperplane in DQ(8, K) and show that every locally singular hyperplane of DQ(8, K) is either singular, the extension of a hexagonal hyperplane in a hex or of the new type.