Published by De Gruyter

Volume 21 Issue 1

Issue of Advances in Geometry

Contents
Journal Overview

Publicly Available January 22, 2021

Frontmatter

Page range: i-iv

Download PDF

September 11, 2019

In simply connected cotangent bundles, exact Lagrangian cobordisms are h-cobordisms

Hiro Lee Tanaka

Page range: 1-4

Abstract

Let Q be a simply connected manifold. We show that every exact Lagrangian cobordism between compact, exact Lagrangians in T * Q is an h -cobordism. This is a corollary of the Abouzaid–Kragh Theorem.

January 25, 2020

A note on norms of signed sums of vectors

Giorgos Chasapis, Nikos Skarmogiannis

Page range: 5-14

Abstract

Improving a result of Hajela, we show for every function f with lim n →∞ f ( n ) = ∞ and f ( n ) = o ( n ) that there exists n 0 = n 0 ( f ) such that for every n ⩾ n 0 and any S ⊆ {–1, 1} n with cardinality | S | ⩽ 2 n / f ( n ) one can find orthonormal vectors x 1 , …, x n ∈ ℝ n satisfying ∥ ε 1x1+⋯ +ε nxn∥ ∞ ⩾ clog⁡ f(n)$\begin{array}{} \displaystyle \|\varepsilon_1x_1+\dots+\varepsilon_nx_n\|_{\infty }\geqslant c\sqrt{\log f(n)} \end{array}$ for all ( ε 1 , …, ε n ) ∈ S . We obtain analogous results in the case where x 1 , …, x n are independent random points uniformly distributed in the Euclidean unit ball B2n$\begin{array}{} \displaystyle B_2^n \end{array}$ or in any symmetric convex body, and the ℓ ∞ n$\begin{array}{} \displaystyle \ell_{\infty }^n \end{array}$-norm is replaced by an arbitrary norm on ℝ n .

January 22, 2021

Ordinary hyperspheres and spherical curves

Aaron Lin, Konrad Swanepoel

Page range: 15-22

Abstract

An ordinary hypersphere of a set of points in real d -space, where no d + 1 points lie on a ( d - 2)-sphere or a ( d - 2)-flat, is a hypersphere (including the degenerate case of a hyperplane) that contains exactly d + 1 points of the set. Similarly, a ( d + 2)- point hypersphere of such a set is one that contains exactly d + 2 points of the set. We find the minimum number of ordinary hyperspheres, solving the d -dimensional spherical analogue of the Dirac–Motzkin conjecture for d ⩾ 3. We also find the maximum number of ( d + 2)-point hyperspheres in even dimensions, solving the d -dimensional spherical analogue of the orchard problem for even d ⩾ 4.

January 22, 2021

Birational models of 𝓜_2,2 arising as moduli of curves with nonspecial divisors

Drew Johnson, Alexander Polishchuk

Page range: 23-43

Abstract

We study birational projective models of 𝓜 2,2 obtained from the moduli space of curves with nonspecial divisors. We describe geometrically which singular curves appear in these models and show that one of them is obtained by blowing down the Weierstrass divisor in the moduli stack of 𝓩-stable curves 𝓜 2,2 (𝓩) defined by Smyth. As a corollary, we prove projectivity of the coarse moduli space M 2,2 (𝓩).

September 11, 2019

The integral polytope group

Florian Funke

Page range: 45-62

Abstract

We show that the Grothendieck group associated to integral polytopes in ℝ n is free-abelian, by providing an explicit basis. Moreover, we identify the involution on this polytope group given by reflection about the origin as a sum of Euler characteristic type. We also compute the kernel of the norm map sending a polytope to its induced seminorm on the dual of ℝ n .

January 22, 2021

Automorphisms of a Clifford-like parallelism

Hans Havlicek, Stefano Pasotti, Silvia Pianta

Page range: 63-73

Abstract

We focus on the description of the automorphism group Γ ∥ of a Clifford-like parallelism ∥ on a 3-dimensional projective double space (ℙ( H F ), ∥ ℓ , ∥ r ) over a quaternion skew field H (of any characteristic). We compare Γ ∥ with the automorphism group Γ ℓ of the left parallelism ∥ ℓ , which is strictly related to Aut( H ). We build up and discuss several examples showing that over certain quaternion skew fields it is possible to choose ∥ in such a way that Γ ∥ is either properly contained in Γ ℓ or coincides with Γ ℓ even though ∥ ≠ ∥ ℓ .

January 22, 2021

The reverse Kakeya problem

Sang Won Bae, Sergio Cabello, Otfried Cheong, Yoonsung Choi, Fabian Stehn, Sang Duk Yoon

Page range: 75-84

Abstract

We prove a generalization of Pál's conjecture from 1921: if a convex shape P can be placed in any orientation inside a convex shape Q in the plane, then P can also be turned continuously through 360° inside Q . We also prove a lower bound of Ω ( m n 2 ) on the number of combinatorially distinct maximal placements of a convex m -gon P in a convex n -gon Q . This matches the upper bound proven by Agarwal et al.

January 22, 2021

K3 polytopes and their quartic surfaces

Gabriele Balletti, Marta Panizzut, Bernd Sturmfels

Page range: 85-98

Abstract

K3 polytopes appear in complements of tropical quartic surfaces. They are dual to regular unimodular central triangulations of reflexive polytopes in the fourth dilation of the standard tetrahedron. Exploring these combinatorial objects, we classify K3 polytopes with up to 30 vertices. Their number is 36 297 333. We study the singular loci of quartic surfaces that tropicalize to K3 polytopes. These surfaces are stable in the sense of Geometric Invariant Theory.

January 22, 2021

Geometrisation of purely hyperbolic representations in PSL2R

Gianluca Faraco

Page range: 99-108

Abstract

Let S be a surface of genus g at least 2. A representation ρ:π1S→PSL2R$ \rho:\pi_1 S\to{\mathrm{PSL}_2\mathbb{R}} $ is said to be purely hyperbolic if its image consists only of hyperbolic elements along with the identity. We may wonder under which conditions such representations arise as the holonomy of a branched hyperbolic structure on S . In this work we characterise them completely, giving necessary and sufficient conditions.

February 17, 2020

Some topics in differential geometry of normed spaces

Vitor Balestro, Horst Martini, Ralph Teixeira

Page range: 109-118

Abstract

For a surface immersed in a three-dimensional space endowed with a smooth norm instead of an inner product, one can define analogous concepts of curvature and metric. With such concepts in mind, various questions immediately appear. The aim of this paper is to propose and answer some of those questions. In this framework we prove several characterizations of minimal surfaces in normed spaces, and respective analogues of some global theorems (e.g., Hadamard-type theorems) are also derived. A result on the curvature of surfaces having constant Minkowskian width is given, and finally we study the ambient metric induced on the surface, proving an extension of the classical Bonnet theorem.

January 22, 2021

A note on non-commutative polytopes and polyhedra

Beatrix Huber, Tim Netzer

Page range: 119-124

Abstract

It is well-known that every polyhedral cone is finitely generated (i.e. polytopal), and vice versa. Surprisingly, the two notions differ almost always for non-commutative versions of such cones, as was recently proved by different authors. In this note we give a direct and constructive proof of the statement. Our proof yields a new and surprising quantitative result: the difference of the two notions can always be seen at the first level of non-commutativity, i.e. for matrices of size 2, independent of dimension and complexity of the initial convex cone.

September 11, 2019

A maximum principle for circle-valued temperatures

Andrew A. Cooper

Page range: 125-132

Abstract

In this note we prove a ‘maximum principle’ for circle-valued solutions of the (time-dependent) heat equation on closed Riemannian manifolds.

February 17, 2020

Extension of the Andersén–Lempert theory: Lie algebras of zero divergence vector fields on complex affine algebraic varieties

Fabrizio Donzelli

Page range: 133-141

Abstract

For a smooth manifold X equipped with a volume form, let 𝓛 0 ( X ) be the Lie algebra of volume preserving smooth vector fields on X . Lichnerowicz proved that the abelianization of 𝓛 0 ( X ) is a finite-dimensional vector space, and that its dimension depends only on the topology of X . In this paper we provide analogous results for some classical examples of non-singular complex affine algebraic varieties with trivial canonical bundle, which include certain algebraic surfaces and linear algebraic groups. The proofs are based on a remarkable result of Grothendieck on the cohomology of affine varieties, and some techniques that were introduced with the purpose of extending the Andersén–Lempert theory, which was originally developed for the complex spaces ℂ n , to the larger class of Stein manifolds that satisfy the density property.

January 22, 2021

Some infinite sequences of canonical covers of degree 2

Nguyen Bin

Page range: 143-148

Abstract

In this note, we construct three new infinite families of surfaces of general type with canonical map of degree 2 onto a surface of general type. For one of these families the canonical system has base points.

About this journal

Objective
Advances in Geometry is a mathematical journal for the publication of original research articles of excellent quality in the area of geometry. Geometry is a field of long standing-tradition and eminent importance. The study of space and spatial patterns is a major mathematical activity; geometric ideas and geometric language permeate all of mathematics.

Topics

Finite geometry
Incidence geometry
Tits buildings
Combinatorial aspects of geometry and geometric combinatorics
Geometric group theory
Geometric aspects of Lie groups and of algebraic groups
Algebraic geometry
Topological geometry
Differential geometry
Convex sets and polytopes
Discrete geometry

Article formats
Research articles

Submit Article

Volume 21 Issue 1

Issue of Advances in Geometry

Frontmatter

In simply connected cotangent bundles, exact Lagrangian cobordisms are h-cobordisms

Abstract

A note on norms of signed sums of vectors

Abstract

Ordinary hyperspheres and spherical curves

Abstract

Birational models of 𝓜2,2 arising as moduli of curves with nonspecial divisors

Abstract

The integral polytope group

Abstract

Automorphisms of a Clifford-like parallelism

Abstract

The reverse Kakeya problem

Abstract

K3 polytopes and their quartic surfaces

Abstract

Geometrisation of purely hyperbolic representations in PSL2R

Abstract

Some topics in differential geometry of normed spaces

Abstract

A note on non-commutative polytopes and polyhedra

Abstract

A maximum principle for circle-valued temperatures

Abstract

Extension of the Andersén–Lempert theory: Lie algebras of zero divergence vector fields on complex affine algebraic varieties

Abstract

Some infinite sequences of canonical covers of degree 2

Abstract

Birational models of 𝓜_2,2 arising as moduli of curves with nonspecial divisors