Related papers: Tropicalization of Positive Grassmannians
We present a constructive proof of Ky Fan's combinatorial lemma concerning labellings of triangulated spheres. Our construction works for triangulations of $S^n$ that contain a flag of hemispheres. As a consequence, we produce a…
We consider a desingularization Gamma of a Richardson variety X_w^v=X_w \cap X^v in the flag variety Fl(n)=GL(n)/B, obtained as a fibre of a projection from a certain Bott-Samelson variety Z. We then construct a basis of the homogeneous…
We compactify the classical moduli variety of compact Riemann surfaces by attaching moduli of (metrized) graphs as boundary. The compactifications do not admit the structure of varieties and patch together to form a big connected moduli…
Let $\varphi\colon\Gamma\to G$ be a homomorphism of groups. We consider factorizations $\Gamma\xrightarrow{f} M\xrightarrow{g} G$ of $\varphi$ such that either $g$ or $f$ are universal normal maps (namely, crossed modules). These two…
This paper supplements [17], showing that categorically the layered theory is the same as the theory of ordered monoids (e.g. the max-plus algebra) used in tropical mathematics. A layered theory is developed in the context of categories,…
The algebraic foundation of tropical polynomial algebra provides the framework for the geometric construction of the supplement and the reversal of tropical varieties, thereby inducing a duality of reduced tropical varieties; for classes of…
The present paper is a sequel to our work on hybrid geometry of curves and their moduli spaces. We introduce a notion of hybrid Laplacian, formulate a hybrid Poisson equation, and give a mathematical meaning to the convergence both of the…
We propose an explicit construction of a weighted generalised Grassmannian. For a weighted Grassmannian (i.e., for series A) we obtain an effective parametrisation of possible $\mathbb{Z}$-gradings on Pl\"{u}cker coordinates, and provide…
We show that the tropicalization of a connected variety over a higher rank valued field is a path connected topological space. This establishes an affirmative answer to a question posed by Banerjee. Higher rank tropical varieties are…
Recently, a theory for the tropicalization of a spherical homogeneous space $G/H$ was developed by Tassos Vogiannou. We extend his ideas to define the tropicalization of a spherical $G/H$-embedding. This generalizes the construction of…
Let $G$ be a simple algebraic group of type $A$ or $D$ defined over $\C$ and $T$ be a maximal torus of $G$. For a dominant coweight $\lambda$ of $G$, the $T$-fixed point subscheme $(\bar{Gr}_G^\lambda)^T$ of the Schubert variety…
We explicitly describe the tropicalization of a cluster variety of finite type C, realizing it as the space of axially symmetric phylogenetic trees. We also find all occurring sign patterns of coordinates, for both the cluster variety and…
This paper lays out a foundation for a theory of supertropical algebraic geometry, relying on commutative $\nu$-algebra. To this end, the paper introduces $\mathfrak{q}$-congruences, carried over $\nu$-semirings, whose distinguished ghost…
For a general linear supergroup $G=GL(m|n)$, we consider a natural isomorphism $\phi: G \to U^-\times G_{ev} \times U^+$, where $G_{ev}$ is the even subsupergroup of $G$, and $U^-$, $U^+$ are appropriate odd unipotent subsupergroups of $G$.…
There are two widely used models for the Grassmannian $\operatorname{Gr}(k,n)$, as the set of equivalence classes of orthogonal matrices $\operatorname{O}(n)/(\operatorname{O}(k) \times \operatorname{O}(n-k))$, and as the set of trace-$k$…
The notion of geometric construction is introduced. This notion allows to compare incidence configurations in the algebraic and tropical plane. We provide an algorithm such that, given a tropical instance of a geometric construction, it…
We define a class of topological A-models on a collection of Riemann surfaces, whose boundaries are sewn together along the seams. The target spaces for the Riemann surfaces are the Grassmanians Gr_{m_i,n} with the common value of n, and…
Given an integral scheme X over a non-archimedean valued field k , we construct acuniversal closed embedding of X into a k-scheme equipped with a model over the field with one element (a generalization of a toric variety). An embedding into…
We introduce a natural generalization of the scattering equations, which connect the space of Mandelstam invariants to that of points on ${\mathbb{CP}^1}$, to higher-dimensional projective spaces $\mathbb{CP}^{k-1}$. The standard, $k=2$…
The tropicalization of a linear space over a non-archimedean field is a tropical linear space. In this paper, we present a method for computing the tropicalization of any lattice over a valuation ring. The resulting tropical semimodule is…