Related papers: Implicitization of surfaces via geometric tropical…
We describe in some details an idea of M. Kontsevich how one can try to find a counterexample to the Hodge conjecture using tropical geometry.
We derive the implicit equations for certain parametric surfaces in three-dimensional projective space termed tensor product surfaces. Our method computes the implicit equation for such a surface based on the knowledge of the syzygies of…
A degeneration of a singular curve on a toric surface, called a tropicalization, was constructed by E. Shustin. He classified the degeneration of 1-cuspidal curves using polyhedral complexes called tropical curves. In this paper, we define…
For certain tropical quartic curves the existing techniques could not predict the lifting behavior of their bitangents over the real numbers. We close this gap by using patchworking techniques. Further, this paper provides an analysis of…
In this article we study holomorphic integrals on tropical plane curves in view of ultradiscretization. We prove that the lattice integrals over tropical curves can be obtained as a certain limit of complex integrals over Riemannian…
Ducros, Hrushovski, and Loeser gave maps from families of archimedean diffrential forms to non-archiemedean (or tropical) ones, which are compatible with integrals on algebraic varieties. In this paper, we introduce slight modifications of…
We present a new algorithmic framework which utilizes tropical geometry and homotopy continuation for solving systems of polynomial equations where some of the polynomials are generic elements in linear subspaces of the polynomial ring.…
The first secant variety of a projective monomial curve is a threefold with an action by a one-dimensional torus. Its tropicalization is a three-dimensional fan with a one-dimensional lineality space, so the tropical threefold is…
We interpret the degrees which arise in Tevelev's study of scattering amplitudes in terms of moduli spaces of Hurwitz covers. Via excess intersection theory, the boundary geometry of the Hurwitz moduli space yields a simple recursion for…
We obtain an explicit expression for the number of ramified coverings of the sphere by the torus with given ramification type for a small number of ramification points, and conjecture this to be true for an arbitrary number of ramification…
We show that the counting of rational curves on a complete toric variety that are in general position to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraic-geometric and relies on…
We use tropical and non-archimedean geometry to study the generic number of solutions of families of polynomial equations over a parameter space $Y$. In particular, we are interested in the choices of parameters for which the generic root…
This work uncovers the tropical analogue for measured laminations of the convex hull construction of decorated Teichmueller theory, namely, it is a study in coordinates of geometric degeneration to a point of Thurston's boundary for…
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…
Given the tropicalization of a complex subvariety of the torus, we define a morphism between the tropical cohomology and the rational cohomology of their respective tropical compactifications. We say that the subvariety of the torus is…
In this paper we study algorithmic aspects of tropical intersection theory. We analyse how divisors and intersection products on tropical cycles can actually be computed using polyhedral geometry. The main focus of this paper is the study…
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,…
Let $g_1, ..., g_k$ be tropical polynomials in $n$ variables with Newton polytopes $P_1, ..., P_k$. We study combinatorial questions on the intersection of the tropical hypersurfaces defined by $g_1, ..., g_k$, such as the $f$-vector, the…
K3 polytopes appear in complements of tropical quartic surfaces. They are dual to regular unimodular central triangulations of reflexive polytopes in the fourth dilation of the standard tetrahedron. Exploring these combinatorial objects, we…
We show how Seifert surfaces, so useful for the understanding of the Alexander polynomial \Delta_L(t), can be generalized in order to study the multivariable Alexander polynomial \Delta_L(t_1,...,t_\mu). In particular, we give an elementary…