Related papers: Self-intersections of surfaces that contain two ci…
This paper is devoted to presenting a new approach to determine the intersection of two quadrics based on the detailed analysis of its projection in the plane (the so called cutcurve) allowing to perform the corresponding lifting correctly.…
In this article, we are concerned with various aspects of arcs on surfaces. In the first part, we deal with topological aspects of arcs and their complements. We use this understanding, in the second part, to construct interesting actions…
We enumerate the singular algebraic curves in a complete linear system on a smooth projective surface. The system must be suitably ample in a rather precise sense. The curves may have up to eight nodes, or a triple point of a given type and…
Let M be a compact connected orientable 3-manifold, with non-empty boundary that contains no 2-spheres. We investigate the existence of two properly embedded disjoint surfaces S_1 and S_2 such that M - (S_1 \cup S_2) is connected. We show…
We uncover some connections between the topology of a complete Riemannian surface M and the minimum number of vertices, i.e., critical points of geodesic curvature, of closed curves in M. In particular we show that the space forms with…
In this note we investigate three new pencils of symmetric surfaces in complex projective three-space. These have degree 6, 8 resp. 12 and are invariant under the action of subgroups of SO(4) containing the Heisenberg group. The pencils of…
Ordinary Coincidence Site Lattices (CSLs) are defined as the intersection of a lattice $\Gamma$ with a rotated copy $R\Gamma$ of itself. They are useful for classifying grain boundaries and have been studied extensively since the mid…
Conjugation spaces are topological spaces equipped with an involution such that their fixed points have the same mod $2$ cohomology (as a graded vector space, a ring, and even an unstable algebra) but with all degrees divided by two,…
We completely classify edge-to-edge tilings of the sphere by congruent quadrilaterals. As part of the classification, we also present a modern version of the classification of edge-to-edge tilings of the sphere by congruent triangles.…
We explore offsets of Cayley ovals, by networking with different kinds of software. Using their specific abilities, algebraic, geometric, dynamic, we conjecture interesting properties of the offsets. For a given progenitor (the given plane…
We track the trajectories of individual horocycles on the modular surface. Our tracking is constructive, and we thus \emph{effectively} establish topological transitivity and even line-transitivity for the horocyclic flow. We also describe…
We consider the following question: Given $n$ lines and $n$ circles in $\mathbb{R}^3$, what is the maximum number of intersection points lying on at least one line and on at least one circle of these families. We prove that if there are no…
In this paper we consider the existence of complete intersection points of type $(a,b,c)$, on the generic degree $d$ surface of $\PP^3$. For any choice of $a, b, c$ we resolve the existence question asymptotically, i.e. for all $d \gg 0$.…
Given a surface $\Sigma$ equipped with a set $P$ of marked points, we consider the triangulations of $\Sigma$ with vertex set $P$. The flip-graph of $\Sigma$ whose vertices are these triangulations, and whose edges correspond to flipping…
Three circles define each of the Brocard points of a triangle. If one adds the three circles through a pair of vertices and the orthocentre one has nine circles. It is described how each of the nine centres of these circles lies at the…
Suppose we have identified three clusters of galaxies as being topological copies of the same object. How does this information constrain the possible models for the shape of our Universe? It is shown here that, if the Universe has flat…
We consider singular holomorphic foliations on compact complex surfaces with invariant rational nodal curve of positive self-intersection. Then, under some assumptions, we list all possible foliations.
A face-transitive surface is a triangulated 2-dimensional manifold whose automorphism group acts transitively on its set of triangles. In this paper, we investigate this class of highly symmetric surface triangulations. We identify seven…
It is well-known that cross caps on surfaces in the Euclidean 3-space can be expressed in Bruce-West's normal form, which is a special local coordinate system centered at the singular point. In this paper, we show a certain kind of…
To a generic configuration of eight points in convex position in the plane, we associate a list consisting of the following information: for all of the 56 conics determined by five of the points, we specify the position of each remaining…