Band 7, Heft 1

Let π be a thick polar space of rank 4 over a field 𝕂 such that the generalised quadrangle Res + (α) of a line α of π which consists of the planes and 3-spaces of π containing α, admits a spread. Let be a line-spread of π with the following property: Let be the set of 3-spaces of π in which induces spreads. For every point Σ of π, the 3-spaces of containing Σ all contain the spread line λ ∈ covering Σ and form a spread of the generalised quadrangle . Then is a generalised quadrangle. The polar spaces Sp 8 (𝕂) and admit such spreads for both finite and infinite fields 𝕂. They are the only finite classical polar spaces admitting such spreads. If π ≅ Sp 8 ( q ), respectively , then Γ ≅ Sp 4 ( q 2 ), respectively H 5 ( q 2 ). If π ≅ Sp 8 (𝕂) for some infinite field 𝕂, then Γ ≅ Sp 4 (ℍ) for some field ℍ. In the infinite case, there exists an example of a spread in H 8 (ℂ) over the complex numbers ℂ with over the quaternions. Dualizing π, the point set of the dual polar space Δ dual to π is a hyperplane of Δ intersecting each symp Σ, i.e. an element of maximal type of Δ, in the set of neighbours of an ovoid of a quad of Σ.

Jørgensen's inequality gives a necessary condition for the discreteness of a non-elementary group of isometries of hyperbolic 3-space. The main idea of the proof may be generalised widely but the statement is quite specialised. Here we give a scheme for restating Jørgensen's inequality for Möbius transformations of a metric space. This unifies many previously published versions of Jørgensen's inequality. We then show how this scheme may be applied by giving a version of Jørgensen's inequality for the octonionic hyperbolic plane.

This article continues the study of multiple blocking sets in PG(2, q ). In [A. Blokhuis, L. Storme, T. Szőnyi, Lacunary polynomials, multiple blocking sets and Baer subplanes. J. London Math. Soc . (2) 60 (1999), 321–332. MR1724814 (2000j:05025) Zbl 0940.51007], using lacunary polynomials, it was proven that t -fold blocking sets of PG(2, q ), q square, t < q ¼ /2, of size smaller than t ( q + 1) + c q q ⅔ , with c q = 2 −⅓ when q is a power of 2 or 3 and c q = 1 otherwise, contain the union of t pairwise disjoint Baer subplanes when t ≥ 2, or a line or a Baer subplane when t = 1. We now combine the method of lacunary polynomials with the use of algebraic curves to improve the known characterization results on multiple blocking sets and to prove a t (mod p ) result on small t -fold blocking sets of PG(2, q = p n ), p prime, n ≥ 1.

We describe an alternative perspective for at least some of the subgroups in Aschbacher's class C 9 . Then we show that under the twisted tensor product embedding of the groups PSp 2 ( q t ) and PSp 4 ( q t ) in orthogonal groups over GF( q ), t ≥ 3, q ≥ 4 even, an intermediate embedding of type C 3 occurs.

We consider a simple condition defining a tetradic set of ovoids in a projective three-space over a finite field. By elementary counting and geometrical methods we establish the properties of a tetradic set and are able to give a purely synthetic construction of the class of generalized quadrangles of order ( s, s 2 ) satisfying Property (G) at a flag. This includes the class of dual flock generalized quadrangles due to Kantor and Payne in the 1980's. We also show that the dual flock generalized quadrangles are characterised by Property (G) at a line.

The farthest point mapping on compact surfaces, associating to each point x of the surface the set of absolute maxima of the intrinsic distance from x , is for some surfaces single-valued and a homeomorphism, while for other surfaces it is not single-valued, and not surjective. These two big classes are not very well understood. For instance it is still unknown whether, say in the convex case, the second class is dense. For a C 2 metric on both surfaces and the space of surfaces, the first class has, however, nonempty interior. We describe various properties of the sets of critical points, and of relative and absolute maxima of distance functions, and find several connections between them. We see for example that, on smooth surfaces homeomorphic to S 2 , a point cannot be critical with respect to more than one other point. Sufficient conditions for a surface to belong to the second class will be formulated and a particular Tannery surface belonging to the boundary of both classes will be presented.

We develop Stiefel–Whitney classes for coherent real analytic sheaves and investigate their applications to analytic cycles on real analytic manifolds.

The Weddle surface is classically known to be a birational (partially desingularized) model of the Kummer surface. In this note we go through its relations with moduli spaces of abelian varieties and of rank two vector bundles on a genus 2 curve. First we construct a moduli space A 2 (3) − parametrizing abelian surfaces with a symmetric theta structure and an odd theta characteristic. Such objects can in fact be seen as Weddle surfaces. We prove that A 2 (3) − is rational. Then, given a genus 2 curve C , we give an interpretation of the Weddle surface as a moduli space of extensions classes (invariant with respect to the hyperelliptic involution) of the canonical sheaf ω of C with ω −1 . This in turn allows to see the Weddle surface as a hyperplane section of the secant variety Sec( C ) of the curve C tricanonically embedded in ℙ 4 .

We present some lower bound estimates for the first eigenvalue of p -Laplace operators on compact Riemannian manifolds with quasi-positive (or nonnegative) Ricci curvature in terms of diameter of the manifolds. For compact manifolds with boundary, we consider the Dirichlet eigenvalue problem with some proper hypothesis.