Related papers: Tritangents and Their Space Sextics
The theta process is a stochastic process of number theoretical origin arising as a scaling limit of quadratic Weyl sums. It can be described in terms of the geodesic flow and an automorphic function on a homogeneous space. This process has…
In this work we are interested in the central configurations of the spatial seven-body problem where six of them are at vertices of two congruents equilateral triangles belong to parallel planes and one triangle is a rotation by the angle…
A degree one del Pezzo surface is the blowup of P^2 at 8 general points. By the classical Cayley-Bacharach Theorem, there is a unique 9th point whose blowup produces a rational elliptic surface with a section. Via this relationship, we…
In algebraic geometry, it is important to provide effective parametrizations for families of curves, both in theory and in practice. In this paper, we present such an effective parametrization for the moduli of genus-$5$ curves that are…
The tilings of the 2-dimensional sphere by congruent triangles have been extensively studied, and the edge-to-edge tilings have been completely classified. However, not much is known about the tilings by other congruent polygons. In this…
We discuss the existence of the angle between two curves in Teichm\"uller spaces and show that, in any infinite dimensional Teichm\"uller space, there exist infinitely many geodesic triangles each of which has the same three vertices and…
A study is made of algebraic curves and surfaces in the flag manifold $\mathbb{F}=SU(3)/T^2$, and their configuration relative to the twistor projection $\pi$ from $\mathbb{F}$ to the complex projective plane $\mathbb{CP}^2$, defined with…
This paper focuses on studying the configuration spaces of graphs realised in $\mathbb C^2$, such that the configuration space is, after normalisation, one dimensional. If this is the case, then the configuration space is, generically, a…
The dualistic structure of statistical manifolds in information geometry yields eight types of geodesic triangles passing through three given points, the triangle vertices. The interior angles of geodesic triangles can sum up to $\pi$ like…
We consider the ring of invariants of n points on the projective line. The space (P^1)^n // PGL_2 is perhaps the first nontrivial example of a Geometry Invariant Theory quotient. The construction depends on the weighting of the n points.…
Curves play a fundamental role across computer graphics, physical simulation, and mathematical visualization, yet most tools for curve design do nothing to prevent crossings or self-intersections. This paper develops efficient algorithms…
Erd\H{o}s and Fishburn studied the maximum number of points in the plane that span $k$ distances and classified these configurations, as an inverse problem of the Erd\H{o}s distinct distances problem. We consider the analogous problem for…
There are two particular $\Theta_6$-graphs - the 6-cycle graphs with a diagonal. We find the planar Tur\'an number of each of them, i.e. the maximum number of edges in a planar graph $G$ of $n$ vertices not containing the given $\Theta_6$…
We report on the implementation of an algorithm for computing the set of all regular triangulations of finitely many points in Euclidean space. This algorithm, which we call down-flip reverse search, can be restricted, e.g., to computing…
We give a classification up to equisingular deformation and compute the fundamental groups of maximizing plane sextics with a type $\mathbf{E}_6$ singular point.
This paper deals with triangulations of the 2-torus with the vertex labeled general octahedral graph $O_4$ which is isomorphic to the complete four-partite graph $K_{2,2,2,2}$; it is known that there exist precisely twelve such…
We compute the class of arithmetic genus two Teichmueller curves in the Picard group of pseudo-Hilbert modular surfaces, distinguished according to their torsion order and spin invariant. As an application, we compute the number of genus…
We study the following question: given a set P of 3d-2 points and an immersed curve G in the real plane R^2, all in general position, how many real rational plane curves of degree d pass through these points and are tangent to this curve.…
We present an algorithm for detecting basepoints of linear series of curves in the plane. Moreover, we give an algorithm for constructing a linear series of curves in the plane for given basepoints. The underlying method of these algorithms…
We prove Manin's conjecture for a del Pezzo surface of degree six which has one singularity of type $\mathbf{A}_2$. Moreover, we achieve a meromorphic continuation and explicit expression of the associated height zeta function.