Related papers: On maximal rational poylhedral fans
Let X be a complex, Gorenstein, Q-factorial, toric Fano variety. We prove two conjectures on the maximal Picard number of X in terms of its dimension and its pseudo-index, and characterize the boundary cases. Equivalently, we determine the…
In a previous work, we stated the conjecture that any semiring homomorphism between the Boolean function semirings on 1-dimensional tropical fans has the property called geometric. In this paper, we show that the conjecture is true. As an…
The aim of this article is to show the existence, and also give an explicit construction, of infinite sets of orthogonal exponentials for certain families of convex polytopes which include simple-rational polytopes and also non simple…
We define and solve the toric version of the symplectic ball packing problem, in the sense of listing all 2n-dimensional symplectic-toric manifolds which admit a perfect packing by balls embedded in a symplectic and torus equivariant…
We conjecture that any connected component $Q$ of the moduli space of triples $(X,E=E_1+\dots+E_n,\Theta)$ where $X$ is a smooth projective variety, $E$ is a normal crossing anti-canonical divisor with a 0-stratum, every $E_i$ is smooth,…
For any lattice congruence of the weak order on permutations, N. Reading proved that gluing together the cones of the braid fan that belong to the same congruence class defines a complete fan, called a quotient fan, and V. Pilaud and F.…
Abstract polytopes are combinatorial objects that generalise geometric objects such as convex polytopes, maps on surfaces and tilings of the space. Chiral polytopes are those abstract polytopes that admit full combinatorial rotational…
We formulate a combinatorial version of the Intersection Hodge Conjecture for projective toric varieties. The conjecture asserts that the subspace of rational Hodge classes in the intersection cohomology $IH^*(X_\Sigma)$ is generated by the…
Topological invariants such as characteristic classes are an important tool to aid in understanding and categorizing the structure and properties of algebraic varieties. In this note we consider the problem of computing a particular…
Given two tropical polynomials $f, g$ on $\mathbb{R}^n$, we provide a characterization for the existence of a factorization $f= h \odot g$ and the construction of $h$. As a ramification of this result we obtain a parallel result for the…
In a previous paper we introduced the notion of a {\it real semigroup} (RS) as an axiomatic framework to study diagonal quadratic forms with arbitrary entries over (commutative, unitary) semi-real rings. Two important classes of RSs were…
We derive a graded character formula for the action of any finite group on the Artinian reduction of the Stanley--Reisner ring of any complete simplicial fan, which is given by an equivariant version of the classical h-polynomial. This…
The purpose of this paper and its sequel (Toric Stacks II) is to introduce and develop a theory of toric stacks which encompasses and extends the notions of toric stacks defined in [Laf02, BCS05, FMN10, Iwa09, Sat12, Tyo12], as well as…
We briefly introduce several problems: (1) a generalization of the convex fair partition conjecture, (2) on non-trivial invariants among polyhedrons that can be formed from the same set of face polygons, (3) two questions on assembling…
We give an explicit projectivization algorithm for smooth complete toric varieties. More precisely, after fixing an ordered lattice basis, every smooth complete fan $\Sigma$ admits a basis-canonical refinement $\widehat{\Sigma}$ that is…
Following DeMeyer, Ford & Miranda [DFM93], we define a topology on a fan by declaring open sets to be its subfans. Then, like Kato [Kat94], we make our fans into monoided spaces by associating a sheaf of monoids to each fan. (Our sheaf of…
We study how geometric properties of tropical convex sets and polytopes, which are of interest in many application areas, manifest themselves in their algebraic structure as modules over the tropical semiring. Our main results establish a…
To quantify the geometric expressivity of transformers, we introduce a tropical geometry framework to characterize their exact spatial partitioning capabilities. By modeling self-attention as a vector-valued tropical rational map, we prove…
We introduce the notion of tropical Lagrangian multi-sections over a fan and study its relation with toric vector bundles. We also introduce a "SYZ-type" construction for toric vector bundles which gives a reinterpretation of Kaneyama's…
We consider complete toric varieties $X$ such that a maximal unipotent subgroup $U$ of the automorphism group $\text{Aut}(X)$ acts on $X$ with an open orbit. It turns out that such varieties can be characterized by several remarkable…