A. Del Fra

Young diagrams and ideals of Pfaffians

Publisher Summary This chapter presents a study of the G -module structure of a component A and presents the decomposition into irreducible modules when the base field K is of char 0. The chapter also presents a complete geometric description of the G -invariant irreducible subvarieties P 2k , where P 2k is the subset of matrices in M n with rank ≤ 2 k . The prime ideal corresponding to P 2k is generated by the Pfaffians of order 2 k + 2 and is referred as a Pfaffian variety. The chapter further describes the ideals of functions vanishing on the Pfaffian varieties P 2k with a prescribed order. It also presents a study of the product of Pfaffian ideals and presents the problems of integral closure.

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.

(0,n)-Sets in a generalized quadrangle

Abstract We study subsets of points in a generalized quadrangle (GQ), in particular, those meeting every line in just n points or in none at all, (i.e: sets of class (0, n )), simply called here (0, n )-sets.

Minimal Blocking Sets in PG( 2 , 8 ) and Maximal Partial Spreads in PG( 3 , 8 )

We prove that PG(2, 8) does not contain minimal blocking sets of size 14. Using this result we prove that 58 is the largest size for a maximal partial spread of PG(3, 8). This supports the conjecture that q2−q+ 2 is the largest size for a maximal partial spread of PG(3, q), q>7.

Diameter bounds for locally partial geometries

We obtain a new upper bound d 1 =[ s /2] − α + 3 for the diameter Δ of a locally partial geometry LpG α ( r, s, t ) of order ( r, s, t ) with α ⩾ 2 (see [10] for α= r = 1). This is attained by an infinite family of LpG 2 (1, s , 1), say D s +2 (see example 1.1). Furthermore, other upper bounds for Δ are obtained, which are often better than the previous one.

C2·c Geometries and Generalized Quadrangles of Order (s − 1, s + 1)

Summary— A family of finite generalized quadrangles, including those of type T2*(O), is characterized in this paper by a simple axiom on geometries belonging to the diagram C2.c. Adding one more axiom, a characterization of 2*(O) is also obtained.

Geometries with bi-affine and bi-linear diagrams

Abstract We consider geometries belonging to the following diagram of rank n ⩾ 4, We prove that when n ⩾ 5, the only simply connected examples for this diagram arise from PG(n,q) by removing a hyperplane and the star of a point. We call these geometries bi-affine geometries. They are of two types, according to whether the point and the hyperplane chosen are incident or not. We also prove that there are just three types of flag-transitive simply connected examples for the rank 4 case of the above diagram, namely the two bi-affine geometries of rank 3 and the (well-known) two-sided extension of PG(2,4) for HS.

Locally partial geometries with different types of residues

Abstract A class of locally partial geometries (L. pGs for short) is constructed where both linear spaces and generalized quadrangles occur as point-residues. We conjecture that this class gives all possible example of L. pGs with that anomaly. We prove this conjecture when plane-residues are projective planes (see section 5). In the general case, we are able to prove the conjecture when there is at least one point with a linear residue, satisfying an additional assumption.

Af * . Af geometries, the Klein quadric and H n q

Abstract An Af ∗ . Af geometry of order q is a residually connected rank three geometry where planes are dual affine planes and stars of points are affine planes of order q . We prove that such a geometry is necessarily obtained from the Klein quadric Q + 5 ( q ) of PG (5, q ) deleting the points of a hyperplane and considering as points the elements of one of the two systems of maximal subspaces of Q = Q + 5 ( q ), as lines the points of Q , and as planes the elements of the other system. The deleted hyperplane is tangent to Q if and only if the Af ∗ . Af geometry obtained satisfies property (PL 1 ) (i.e. there is a unique plane on every point-line antiflag). When (PL 1 ) is satisfied, some generalizations are obtained for L ∗ . L (resp. N ∗ . L ) geometries (i.e. residually connected rank three geometries where planes are dual linear spaces (resp. dual nets) and stars of points are linear spaces). In particular, this yields a characterization of H n q and T ∗ 2 ( K ), where K = AG (2, q ), in the context of rank 3 partial geometries. Furthermore, it leads to some classification results for other rank n ⩾3 diagrams related to what we call the rank n H n q ( n ⩾3, see Examples 5.1 and 5.4 for the definition).

Flag-transitive C 2 . L n geometries

Abstract In this paper we prove that the only locally finite, thick flag-transitive C n . L geometries with n ⩾ 3 are truncations of polar spaces. We recall that for n = 2 an example of thick flag-transitive geometry which is not a truncated polar space has been given by Ronan (1980, 1986). Moreover, we prove that no flag-transitive thick C 2 . A f . A n − 2 . L geometry exists with classical generalized quadrangles as lower residues of elements of type 2, except possibly when q = 3 or 4. However there are examples of flag-transitive thick C 2 . A f . A n − 2 . L geometries where the lower residue of a plane is isomorphic to the generalized quadrangle dual of T ∗ (O) .