Geometriae Dedicata

We characterize a class of linear spaces by the property that through any point outside two disjoint, but non-parallel lines there is at most one transversal.

We show that for X a proper $$\textrm{CAT}(-1)$$ space there is a maximal open subset of the horofunction compactification of $$X\times X$$ , with respect to the maximum metric, that compactifies the diagonal action of an infinite quasi-convex group of the isometries of X. We also consider the product action of two quasi-convex representations of an infinite hyperbolic group on the product of two different proper $$\textrm{CAT}(-1)$$ spaces.

Let X be a smooth projective curve of genus g ≥ 2 over the complex numbers. A holomorphic triple $${(E_1, E_2, \phi)}$$ on X consists of two holomorphic vector bundles E 1 and E 2 over X and a holomorphic map $${\phi \colon E_{2}\to E_{1}}$$ . There is a concept of stability for triples which depends on a real parameter σ. In this paper, we determine the Hodge polynomials of the moduli spaces of σ-stable triples with rk(E 1) = 3, rk(E 2) = 1, using the theory of mixed Hodge structures. This gives in particular the Poincaré polynomials of these moduli spaces. As a byproduct, we recover the Hodge polynomial of the moduli space of odd degree rank 3 stable vector bundles.

The aim of this paper is to give a geometric interpretation of the continued fraction expansion in the field $$\hat K = \mathbb{F}_q ((X^{ - 1} ))$$ of formal Laurent series in X −1 over $$\mathbb{F}_q $$ , in terms of the action of the modular group $${\text{SL}}_{\text{2}} (\mathbb{F}_q [X])$$ on the Bruhat–Tits tree of $${\text{SL}}_{\text{2}} (\hat K)$$ , and to deduce from it some corollaries for the diophantine approximation of formal Laurent series in X −1 by rational fractions in X.

Let A and B be disjoint sets of points in PG(2, q) the Desarguesian projective plane of order q, with |A|≥q, |B|=q+1, such that each line through a point of A meets B (just once). Then B is a line.

A collection $$ \Delta $$ of simple closed curves on an orientable surface is an algebraic k-system if the algebraic intersection number $$ \langle \alpha , \beta \rangle $$ is equal to k in absolute value for every $$ \alpha , \beta \in \Delta $$ . Generalizing a theorem of Malestein et al. (Geom Dedicata 168(1):221–233, 2014. doi:10.1007/s10711-012-9827-9) we compute that the maximum size of an algebraic k-system of curves on a surface of genus g is $$2g+1$$ when $$g\ge 3$$ or k is odd, and 2g otherwise. To illustrate the tightness in our assumptions, we present a construction of curves pairwise geometrically intersecting twice whose size grows as $$g^2$$ .

Given a regular –-hermitian form on an n-dimensional vector space V over a commutative field K of characteristic ≠ 2 ( $$n \in \mathbb{N} $$ ). Call an element σ of the unitary group a quasi-involution if σ is a product of commuting quasi-symmetries (a quasi-symmetry is a unitary transformation with a regular (n−1)-dimensional fixed space). In the special case of an orthogonal group every quasi-involution is an involution. Result: every unitary element is a product of five quasi-involutions. If K is algebraically closed then three quasi-involutions suffice.