Related papers: The surface of a lattice polytope
We present methods to construct interesting surfaces of general type via $\mathbb{Q}$-Gorenstein smoothing of a singular surface obtained from an elliptic surface. By applying our methods to special Enriques surfaces, we construct new…
In this paper we investigate the number of integer points lying in dilations of lattice path matroid polytopes. We give a characterization of such points as polygonal paths in the diagram of the lattice path matroid. Furthermore, we prove…
The formula for the area of a rhumb polygon, a polygon whose edges are rhumb lines on an ellipsoid of revolution, is derived and a method is given for computing the area accurately. This paper also points out that standard methods for…
In a d-simplex every facet is a (d-1)-simplex. We consider as generalized simplices other combinatorial classes of polytopes, all of whose facets are in the class. Cubes and multiplexes are two such classes of generalized simplices. In this…
We derive a formally simple approximate analytical solution to the Poisson-Boltzmann equation for the spherical system via a geometric mapping. Its regime of applicability in the parameter space of the spherical radius and the surface…
The Ehrhart quasipolynomial of a rational polytope $P$ encodes the number of integer lattice points in dilates of $P$, and the $h^*$-polynomial of $P$ is the numerator of the accompanying generating function. We provide two decomposition…
A spherical polyhedron surface is a triangulated surface obtained by isometric gluing of spherical triangles. For instance, the boundary of a generic convex polytope in the 3-sphere is a spherical polyhedron surface. This paper investigates…
We prove a sharp upper bound on the number of boundary lattice points of a rational polygon in terms of its denominator and the number of interior lattice points, generalizing Scott's inequality. We then give sharp lower and upper bounds on…
We give cohomological criteria for logarithmic good reduction of elliptic surfaces up to modification. Along the way, we prove several more general results about such surfaces in positive characteristic, as well as about log smooth…
We study the harmonic polytope, which arose in Ardila, Denham, and Huh's work on the Lagrangian geometry of matroids. We describe its combinatorial structure, showing that it is a $(2n-2)$-dimensional polytope with…
The simplest version of the Spin-polynomial invariants of the underlying differentiable structures of algebraic surfaces were considered and the simplest arguments were used in order to distinguish the underlying smooth structures of…
We study the extension complexity of polytopes with few vertices or facets. On the one hand, we provide a complete classification of $d$-polytopes with at most $d+4$ vertices according to their extension complexity: Out of the…
Integrals related to the surface area of arbitrary ellipsoids are derived, evaluated, and compared with each other and existing integrals found in the literature. We clarify the literature on the ellipsoid area problem, which dates back to…
This paper is to study the Ehrhart function $L(P,t)$ of a rational $n$-polytope $P$, defined as the number of lattice points of dilated polytopes $tP$ with real numbers $t\geq 0$. It turns out that $L(P,t)$ is a quasi-polynomial of real…
The classical result of Cauchy's surface area formula states that the surface area of the boundary $\partial K=\Sigma$ of any $n$-dimensional convex body in the $n$-dimensional Euclidean space $\mathbb{R}^n$ can be obtained by the average…
Given a set $S \subseteq \mathbb{R}^d$, a hollow polytope has vertices in $S$ but contains no other point of $S$ in its interior. We prove upper and lower bounds on the maximum number of vertices of hollow polytopes whose facets are…
A geometrization of a Kronecker $h$-regular multi-time Lagrangian function with partial derivatives of order one is described, in the sense of d-connections, d-torsions and d-curvatures.
One of the main questions in the theory of normal surface singularities is to understand the relations between their geometry and topology. The lattice cohomology is an important tool in the study of topological properties of a plumbed…
The Hessian of a general cubic surface is a nodal quartic surface, hence its desingularisation is a K3 surface. We determine the transcendental lattice of the Hessian K3 surface for various cubic surfaces (with nodes and/or Eckardt points…
The lattice definition of the two-dimensional topological quantum field theory [Fukuma, {\em et al}, Commun.~Math.~Phys.\ {\bf 161}, 157 (1994)] is generalized to arbitrary (not necessarily orientable) compact surfaces. It is shown that…