Related papers: $N$-Division Points of Hypocycloids

This note (which makes no claim to novelty) presents a proof of the separable rational connectedness of smooth cubic hypersurfaces, in any characteristic, by showing how to explicitly construct very free curves (of degree 3) on them. -----…

For every integer $\ell$, we construct a cubic 3-vertex-connected planar bipartite graph $G$ with $O(\ell^3)$ vertices such that there is no planar straight-line drawing of $G$ whose vertices all lie on $\ell$ lines. This strengthens…

Motivated by questions from Ehrhart theory, we present new results on discrete equidecomposability. Two rational polygons $P$ and $Q$ are said to be discretely equidecomposable if there exists a piecewise affine-unimodular bijection…

Let $M$ be a non-compact hyperbolic $3$-manifold with finite volume and totally geodesic boundary components. By subdividing mixed ideal polyhedral decompositions of $M$, under some certain topological conditions, we prove that $M$ has an…

Euclidean geometry consists of straightedge-and-compass constructions and reasoning about the results of those constructions. We show that Euclidean geometry can be developed using only intuitionistic logic. We consider three versions of…

We describe geometrically and algebraically the set of unattainable points for the Rational Hermite Interpolation Problem (i.e. those points where the problem does not have a solution). We show that this set is a union of equidimensional…

We extend non-emtpyness and irreducibility of Hassett divisors to the moduli spaces of $M$-polarizable cubic fourfolds for higher rank lattices $M$, which in turn provides a systematic approach for describing the irreducible components of…

Using Voisin's method we prove that a very general hypersurface of degree at least 4 in complex projective space of dimension 6, 7, 8 or 9 is not stably rational and so, in particular, not rational. We obtain the same conclusion for the…

All rational parametric curves with prescribed polynomial tangent direction form a vector space. Via tangent directions with rational norm, this includes the important case of rational Pythagorean hodograph curves. We study vector subspaces…

We construct examples of nonresolvable generalized $n$-manifolds, $n\geq 6$, with arbitrary resolution obstruction, homotopy equivalent to any simply connected, closed $n$-manifold. We further investigate the structure of generalized…

Working over the field of order 2 we consider those complete caps (maximal sets of points with no three collinear) which are disjoint from some codimension 2 subspace of projective space. We derive restrictive conditions which such a cap…

Using a construction that builds a monoid from a monoid action, this paper exhibits an example of a direct product of monoids that admits a prefix-closed regular cross-section, but one of whose factors does not admit a regular…

We compute some numerical invariants of the lines on hyperplane sections of a smooth cubic threefold over complex numbers. We also prove that for any smooth hypersurface $X\subset \mathbb P^{n+1}$ of degree $d$ over an algebraically closed…

We compute the facets of the effective and movable cones of divisors on the blow-up of $\mathbb{P}^n$ at $n+3$ points in general position. Given any linear system of hypersurfaces of $\mathbb{P}^n$ based at $n+3$ multiple points in general…

For a Euclidean building $X$ of type $A_{2}$, we classify the 0-dimensional subbuildings $A$ of $\partial_{T}X$ that occur as the asymptotic boundary of closed convex subsets. In particular, we show that triviality of the holonomy of a…

The aim of this paper is to explain how, through the work of a number of people, some algebraic structures related to groupoids have yielded algebraic descriptions of homotopy n-types. Further, these descriptions are explicit, and in some…

An $n\times n\times\dots\times n$ hypercube is made from $n^d$ unit hypercubes. Two unit hypercubes are neighbours if they share a $(d-1)$-dimensional face. In each step of a dismantling process, we remove a unit hypercube that has…

We introduce bridge trisections of knotted surfaces in the four-sphere. This description is inspired by the work of Gay and Kirby on trisections of four-manifolds and extends the classical concept of bridge splittings of links in the…

We construct a sequence of convex polyhedra on n vertices with the property that, as n -> infinity, the fraction of its edge unfoldings that avoid overlap approaches 0, and so the fraction that overlap approaches 1. Nevertheless, each does…

We study supersolvable line arrangements in ${\mathbb P}^2$ over the reals and over the complex numbers, as the first step toward a combinatorial classification. Our main results show that a nontrivial (i.e., not a pencil or near pencil)…