Volume 7, Issue 2

Given a convex polygon P in the projective plane we can form a finite “grid” of points by taking the pairwise intersections of the lines extending the edges of P . When P is a Poncelet polygon we show that this grid is contained in a finite union of ellipses and hyperbolas and derive other related geometric information about the grid.

A convex polytope admits a Coxeter decomposition if it is tiled by finitely many Coxeter polytopes such that any two tiles having a common facet are symmetric with respect to this facet. In this paper, we classify all Coxeter decompositions of compact hyperbolic Coxeter n -polytopes with n + 2 facets. Furthermore, going out from Schläfli‘s reduction formula for simplices we construct in a purely combinatorial way a volume formula for arbitrary polytopes and compute the volumes of all compact Coxeter polytopes in ℍ 4 which are products of simplices.

This is the second paper in a series of three dealing with the classification of the dense near octagons of order (3, t ). In [B. De Bruyn, Dense near octagons with four points on every line, I. Ann. Combin ., to appear.], we determined all dense near octagons of order (3, t ) with a big hex. In the present paper, we will obtain a partial classification of the valuations of the dense near hexagons of order (3, t ). We shall use this classification to obtain some classification results regarding dense near polygons of order (3, t ) containing exceptional hexes. The classification of the valuations will be used in [B. De Bruyn, Dense near octagons with four points on every line, III. Preprint.] to show that almost all dense near octagons of order (3, t ) have a big hex.

In this paper we give conditions for the integrability of almost complex structures calibrated by symplectic forms. We show that in the symplectic case the Newlander–Nirenberg theorem reduces to and we give integrability conditions in terms of the curvature and the Hermitian curvature of the induced metric.

Making suitable generalizations of known results we prove some general facts about Gaussian maps. These facts are then used, in the second part of the article, to give a set of conditions that insure the surjectivity of Gaussian maps for curves on Enriques surfaces. To do this we also solve a problem of independent interest: a tetragonal curve of genus g ≥ 7 lying on an Enriques surface and general in its linear system, cannot be, in its canonical embedding, a quadric section of a surface of degree g − 1 in ℙ g −1 .

We prove that the Lie geometry of a locally compact connected Laguerre space forms a compact connected generalized quadrangle. Moreover, if the geometric dimension of such a Laguerre space is at least three, this generalized quadrangle is isomorphic to a generalized quadrangle of Tits type; all compact connected generalized quadrangles of Tits type arise from Laguerre spaces.

In this paper we study the geometric meaning of the roots of the Steiner polynomial in the 3-dimensional space. We give a complete characterization of the convex bodies in ℝ 3 depending on the type of roots of their Steiner polynomials. Furthermore, we show that these roots are also related to the famous Blaschke problem and the Teissier conjecture.

A circular surface is a one-parameter family of standard circles in ℝ 3 . In this paper some correspondences between the properties of circular surfaces and those of classical ruled surfaces are investigated. Singularities of circular surfaces are also studied.