Related papers: Dynamical Tropicalisation
In this article we define a natural tropicalization procedure for closed subsets of log-regular varieties in the case of constant coefficients and study its basic properties. This framework allows us to generalize some of Tevelev's results…
Much of dynamical algebraic combinatorics focuses on global dynamical systems defined via maps that are compositions of local toggle operators. The second author and Roby studied such maps that result from toggling independent sets of a…
A general problem in complex cobordism theory is to find useful representatives for cobordism classes. One particularly convenient class of complex manifolds consists of smooth projective toric varieties. The bijective correspondence…
For a complex hypersurface of dimension $d \geq 1$ in a toric variety, we construct lifts of tropical $(p, q)$-cycles with $p+q=d$ in the associated tropical hypersurface. The tropical cycles we consider are described by Minkowski weights,…
In this paper, we study tropicalisations of families of curves with a singularity in a fixed point. The tropicalisation of such a family is a linear tropical variety. We describe its maximal dimensional cones using results about linear…
This note presents two observations which have in common that they lie at the boundary of toric geometry. The first one because it concerns the deformation of affine toric varieties into non toric germs in order to understand how to avoid…
Tropical algebraic geometry is the geometry of the tropical semiring $(\mathbb{R},\min,+)$. Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on…
Let $K$ be a real closed field with a nontrivial non-archimedean absolute value. We study a refined version of the tropicalization map, which we call real tropicalization map, that takes into account the signs on $K$. We study images of…
Under some mild assumptions, an orientation-preserving branched covering map of marked $2$-spheres induces a pullback map between the corresponding Teichm\"uller spaces. By analyzing the associated pushforward operator acting on integrable…
We develop a theory of umkehr maps for twisted generalized homology theories. In this theory, interesting umkehr maps, including generalizations of important classical ones, are induced by cartesian morphisms of a certain category opfibred…
Tropical Geometry and Mathematical Morphology share the same max-plus and min-plus semiring arithmetic and matrix algebra. In this chapter we summarize some of their main ideas and common (geometric and algebraic) structure, generalize and…
Given a rational monomial map, we consider the question of finding a toric variety on which it is algebraically stable. We give conditions for when such variety does or does not exist. We also obtain several precise estimates of the degree…
We define and investigate the tropical Prym varieties associated to unramified Galois cyclic covers of tropical curves (or equivalently metric graphs) $\tilde{\Gamma}\to \Gamma$. Our approach here is to study the tropical Prym varieties…
The purpose of this paper is to provide a new account of multiplicity for finite morphisms between smooth projective varieties. Traditionally, this has been defined using commutative algebra in terms of the length of integral ring…
We derive an expression for the local transverse polarization of a boost-invariant expanding system of massive particles, which involves a set of dynamical spin moments. Starting from spin kinetic theory, we obtain a closed set of equations…
We present two effective tools for computing the positive tropicalization of algebraic varieties. First, we outline conditions under which the initial ideal can be used to compute the positive tropicalization, offering a real analogue to…
We present and expand some existing results on the Zariski closure of cyclic groups and semigroups of matrices. We show that, with the exclusion of isolated points, their irreducible components are toric varieties. Additionally, we…
Abstractly, tropical hyperelliptic curves are metric graphs that admit a two-to-one harmonic morphism to a tree. They also appear as embedded tropical curves in the plane arising from triangulations of polygons with all interior lattice…
We consider a class of area-preserving, piecewise affine maps on the 2-sphere. These maps encode degenerating families of K3 surface automorphisms and are profitably studied using techniques from tropical and Berkovich geometries.
This is a tutorial on some aspects of toric varieties related to their potential use in geometric modeling. We discuss projective toric varieties and their ideals, as well as real toric varieties and the algebraic moment map. In particular,…