Related papers: Inherited conics in Hall planes
The geometry of the Heisenberg group acting on the plane arises naturally in geometric topology as a degeneration of the familiar spaces $\mathbb{S}^2,\mathbb{H}^2$ and $\mathbb{E}^2$ via conjugacy limit as defined by Cooper, Danciger, and…
Cutting a polytope is a very natural way to produce new classes of interesting polytopes. Moreover, it has been very enlightening to explore which algebraic and combinatorial properties of the orignial polytope are hereditary to its…
Artin's conjecture is established for all forms that can be realised as a diagonal form on an hyperplane.
The inscribed angle theorem, a famous result about the angle subtended by a chord within a circle, is well known and commonly taught in school curricula. In this paper, we present a generalisation of this result (and other related circle…
As another application of the degeneration methods of [V3], we count the number of irreducible degree $d$ geometric genus $g$ plane curves, with fixed multiple points on a conic $E$, not containing $E$, through an appropriate number of…
This paper deals with non-Archimedean representations of punctured surface groups in PGL(3), associated actions on Euclidean buildings (of type A2), and degenerations of real convex projective structures on surfaces. The main result is…
In this work, we provide some novel results that establish both the existence of Henig global proper efficient points and their density in the efficient set for vector optimization problems in arbitrary normed spaces. Our results do not…
In the present paper we prove that Hall polynomial exists for each triple of decomposition sequences which parameterize isomorphism classes of coherent sheaves of a domestic weighted projective line $\mathbb X$ over finite fields. These…
A regularization procedure developed in [1] for the integral curvature invariants on manifolds with conical singularities is generalized to the case of squashed cones. In general, the squashed conical singularities do not have rotational…
We give an alternate formulation of pseudo-coherence over an arbitrary derived stack X. The full subcategory of pseudo-coherent objects forms a stable sub-infinity-category of the derived category associated to X. Using relative…
Let $A$ be a tame hereditary algebra over a finite field $k$ with $q$ elements, and ${\bar{A}}$ be the duplicated algebra of $A$. In this paper, we investigate the structure of Ringel-Hall algebra $\mathscr{H} (\bar{A})$ and of the…
In this note we present examples of $K(\pi,1)$-arrangements which admit a restriction which fails to be $K(\pi,1)$. This shows that asphericity is not hereditary among hyperplane arrangements.
We collect some results in combinatorial geometry that follow from an inequality of Langer in algebraic geometry. Langer's inequality gives a lower bound on the number of incidences between a point set and its spanned lines, and was…
A set of n points in the plane which are not all collinear defines at least n distinct lines. Chen and Chv\'atal conjectured in 2008 that a similar result can be achieved in the broader context of finite metric spaces. This conjecture…
Let E be a stable rank 2 vector bundle on a smooth quadric threefold Q in the projective 4-space P. We show that the hyperplanes H in P for which the restriction of E to the hyperplane section of Q by H is not stable form, in general, a…
An old theorem, due to Graustein, asserts that the average curvature of a plane oval is attained at least at four points. We present a proof by way of wave propagation and extend this result to the spherical and hyperbolic geometries - in…
We show in this article how Girard Desargues, in his well known text on conics, the \textit{Brouillon Project,} manages to use Menelaos' theorem with some awesome virtuosity. To this end, we propose a detailed analysis of his…
Nondegenerate plane congruences in the four-dimensional complex projective space with degenerate general focal conic are classified by using the focal method due to Corrado Segre.
In PG(2,32) the following two results are proven by a computer aided search. (i) Uniqueness of hyperfocused 12-arcs, up to projectivities; (ii) Non-existence of hyperfocused 14-arcs. The existence problem for hyperfocused 16-arcs remains…
The Hall coefficient is measured in YBa$_2$Cu$_3$O$_y$ untwinned single crystals for a wide range of doping. We show that the Hall conductivity and the Hall angle of the {\it CuO$_2$ planes} in YBa$_2$Cu$_3$O$_y$ can be extracted from…