Volume 25, Issue 4

Given a pair of translation surfaces it is very difficult to determine whether they are supported on the same algebraic curve. In fact, there are very few examples of such pairs. In this note we present infinitely many examples of finite collections of translation surfaces supported on the same algebraic curve. The underlying curves are hyperelliptic curves with many automorphisms. For each curve X , the automorphism group Aut( X ) acts on the space of holomorphic 1-forms. We present a translation surface corresponding to each of the eigenforms of the automorphism of maximal order.

An embedded twisted paper cylinder of aspect ratio λ is a smooth isometric embedding of a flat λ × 1 cylinder into ℝ 3 such that the images of the boundary components are linked. We prove that for such an object to exist we must have λ > 2 and that this bound is sharp. We also show that any sequence of examples having aspect ratio converging to 2 must converge to a (non-smooth) 4-fold wrapping of a right-angled isosceles triangle.

Let V be a general three-dimensional quartic in the complex projective space ℙ 4 such that the only singularity of V is a double point of rank 3. We prove that V is a birationally rigid variety. Its group of birational self-maps is, up to the finite subgroup of biregular automorphisms, a free product of 25 cyclic groups of order 2. It follows that the complement to the set of birationally rigid factorial quartics with terminal singularities is of codimension at least 3 in the natural parameter space.

In this paper we examine the topology of Brill–Noether varieties associated to real trigonal curves. More precisely, we aim to count the connected components of the real locus of the varieties parametrizing linear systems of degree d and dimension at least r . We do this count when the relations m = g − d + r − 1 ≤ d − 2 r − 1 are satisfied, where m is the Maroni invariant and g is the genus of the curve.

We introduce the Hoffman–Singleton manifold based on a particular subgraph of the Hoffman–Singleton graph. This manifold is motivated in a combinatorial fashion, and it is defined rigorously in geometric terms. We also present a few geometric properties of this manifold.

The inscribed square problem asks if every Jordan curve contains the four vertices of a non-degenerate Euclidean square. This question was posed by Otto Toeplitz in 1911 and remains open to this date. It is known that, with the exception of the arc and the 3-star, every topological copy of a locally connected plane continuum in ℝ 2 contains the four vertices of at least one Euclidean rectangle; see [13]. Given ε > 0, a hyperbolic ε -rectangle is a hyperbolic quadrilateral such that its inner angles sum up to more than 2 π − ε , and such that its diagonals share the midpoint and have the same hyperbolic length. A tree T quasi-inscribes rectangles if for every ε > 0, any embedding of T in the hyperbolic plane admits an inscribed ε -rectangle. In this paper, we classify trees that quasi-inscribe rectangles. We also show that any Jordan curve J in the hyperbolic plane admits at least one equilateral hyperbolic triangle whose area is less than π − K , where K is a positive constant that depends only on the diameter of J .

Ramification points arise from singularities along solution paths of a homotopy. This paper considers ramification points of homotopies, elucidating the total number of ramification points and providing general theory regarding the properties of the set of ramification points over the same branch point. The general approach utilized in this paper is to view homotopies as lines in the parameter spaces of families of polynomial systems on a projective manifold. With this approach, the number of singularities of systems parameterized by pencils is computed under broad conditions. General conditions are given for when the singularities of the systems parameterized by a line in a space of polynomial systems have multiplicity two. General conditions are also given for there to be at most one singularity in the solution set of any system parameterized by such a line. Several examples are included to demonstrate the theoretical results.

We characterize q -ample Ulrich bundles on a variety X ⊆ ℙ N with respect to ( q +1)-dimensional linear spaces contained in X .

We give a classification of rotational cmc surfaces in non-Euclidean space forms in terms of explicit parametrizations using Jacobi elliptic functions. Our method hinges on a Lie sphere geometric description of rotational linear Weingarten surfaces and can thus be applied to a more general class of surfaces. As another application of this framework, we give explicit parametrizations of a class of rotational constant harmonic mean curvature surfaces in hyperbolic space. In doing so, we close the last gaps in the classification of all channel linear Weingarten surfaces in space forms, started in [24].

We show that a simply connected stable plane with connected lines is isomorphic to an open subplane of a classical projective plane (i.e., a plane over the real or complex numbers, the quaternions or the octonions) if it has that property locally. We also give several examples indicating that our hypotheses cannot be relaxed much further.

In this paper we prove that there is a flock in every locally compact 2-dimensional Laguerre plane, i.e., a set of disjoint circles which covers the point space. We then determine the homeomorphism type of these flocks and use the relation between circle planes and generalised quadrangles to obtain analogous results about generalised quadrangles, Möbius planes and Minkowski planes.