Volume 24, Issue 2

The Ehrhart polynomial ehr P ( n ) of a lattice polytope P counts the number of integer points in the n -th dilate of P . The f * -vector of P , introduced by Felix Breuer in 2012, is the vector of coefficients of ehr P ( n ) with respect to the binomial coefficient basis {n− 10,n− 11,… ,n− 1d}, $\begin{array}{} \bigl\{\binom{n-1}{0},\binom{n-1}{1},\dots,\binom{n-1}{d}\bigr\}, \end{array}$ where d = dim P . Similarly to h/h * -vectors, the f * -vector of P coincides with the f -vector of its unimodular triangulations (if they exist). We present several inequalities that hold among the coefficients of f * -vectors of lattice polytopes. These inequalities resemble striking similarities with existing inequalities for the coefficients of f -vectors of simplicial polytopes; e.g., the first half of the f * -coefficients increases and the last quarter decreases. Even though f * -vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. We also show that for any polytope with a given Ehrhart h * -vector, there is a polytope with the same h * -vector whose f * -vector is unimodal.

Based on Zong’s work [26] on translative packing densities of 3-dimensional convex bodies, we present a local method to estimate the density θ t ( C 3 ) of the densest translative covering of an octahedron. As a consequence we prove that θ t ( C 3 ) ≥ 1 + 6.6 × 10 –8 , which is the first non-trivial lower bound for this density.

We present two applications of Hao’s proof of the Weak Bounded Negativity Conjecture . First, we address the so-called Weighted Bounded Negativity Conjecture and we prove that all but finitely many reduced and irreducible curves C on the blow-up of ℙ 2 at n points satisfy the inequality C2≥ min{− 112n(C.L+27),− 2}, $\begin{array}{} \displaystyle C^2 \ge \min \bigl\{-\frac{1}{12} n (C.L +27), -2 \bigr\}, \end{array}$ where L is the pull-back of a line. Next, we turn to the widely open conjecture that the canonical degree C . K X of an integral curve on a smooth projective surface X is bounded from above by an expression of the form A ( g − 1) + B , where g is the geometric genus of C and A , B are constants depending only on X . We prove that this conjecture holds with A = − 1 under the assumptions h 0 ( X , − K X ) = 0 and h 0 ( X , 2 K X + C ) = 0.

We demonstrate the construction of Poisson structures via Lie algebroids on moduli spaces of twisted stable Higgs bundles over stacky curves. The construction provides new examples of Poisson structures on such moduli spaces. Special attention is paid to moduli spaces of parabolic Higgs bundles over a root stack.

A Shioda–Inose structure is a geometric construction which associates to an Abelian surface a projective K3 surface in such a way that their transcendental lattices are isometric. This geometric construction was described by Morrison by considering special symplectic involutions on the K3 surfaces. After Morrison several authors provided explicit examples. The aim of this paper is to generalize Morrison’s results and some of the known examples to an analogous geometric construction involving not involutions, but order 3 automorphisms. Therefore, we define generalized Shioda–Inose structures of order 3, we identify the K3 surfaces and the Abelian surfaces which appear in these structures and we provide explicit examples.

For a ∈ R≥ 0n $\begin{array}{} \displaystyle \mathbb{R}_{\geq 0}^{n} \end{array}$, the Tesler polytope Tes n ( a ) is the set of upper triangular matrices with non-negative entries whose hook sum vector is a . We first give a different proof of the known fact that for every fixed a 0 ∈ R> -->0n $\begin{array}{} \displaystyle \mathbb{R}_{ \gt 0}^{n} \end{array}$, all the Tesler polytopes Tes n ( a ) are deformations of Tes n ( a 0 ). We then calculate the deformation cone of Tes n ( a 0 ). In the process, we also show that any deformation of Tes n ( a 0 ) is a translation of a Tesler polytope. Lastly, we consider a larger family of polytopes called flow polytopes which contains the family of Tesler polytopes and chracterize the flow polytopes which are deformations of Tes n ( a 0 ).

On a 7-manifold with a G 2 -structure, we study conformal symmetries — which are vector fields whose flow generate conformal transformations of the G 2 -structure. In particular, we focus on compact 7-manifolds and the condition that the Lee form of the G 2 -structure is closed. Among other observations, we show that conformal symmetries are determined within a conformal class of the G 2 -structure by the symmetries of a unique (up to homothety) G 2 -structure whose Lee form is harmonic. On a related note, we also demonstrate that symmetries are split along fibrations when the Lee vector field is itself a symmetry.

Let n ≥ 3 and let K ⊂ ℝ n be a convex body. A point p ∈ int K is said to be a Larman point of K if for every hyperplane Π passing through p , the section Π ∩ K has an ( n – 2)-plane of symmetry. If p is a Larman point of K and for every section Π ∩ K , p is in the corresponding ( n – 2)-plane of symmetry, then we call p a revolution point of K . We conjecture that if K contains a Larman point which is not a revolution point, then K is either an ellipsoid or a body of revolution. This generalizes a conjecture of Bezdek for n = 3. We prove several results related to the conjecture for strictly convex origin symmetric bodies. Namely, if K ⊂ ℝ n is a strictly convex origin symmetric body that contains a revolution point p which is not the origin, then K is a body of revolution. This generalizes the False Axis of Revolution Theorem in [7]. We also show that if p is a Larman point of K ⊂ ℝ 3 and there exists a line L such that p ∉ L and, for every plane Π passing through p , the line of symmetry of the section Π ∩ K intersects L , then K is a body of revolution (in some cases, K is a sphere). We obtain a similar result for projections of K . Additionally, for K ⊂ ℝ n with n ≥ 4, we show that if every hyperplane section or projection of K is a body of revolution and K has a unique diameter D , then K is a body of revolution with axis D .

We prove that if the geodesic flow on a surface has an integral which is fractional-linear in momenta, then the dimension of the space of such integrals is either 3 or 5, the latter case corresponding to constant gaussian curvature. We give also a geometric criterion for the existence of fractional-linear integrals: such an integral exists if and only if the surface carries a geodesic 4-web with constant cross-ratio of the four directions tangent to the web leaves.

In this article we look at the geometric structure of the feet of an orthogonal Buekenhout–Metz unital 𝓤 in PG(2, q 2 ). We show that the feet of each point form a set of type (0, 1, 2, 4). Further, we discuss the structure of any 4-secants, and determine exactly when the feet form an arc.