The Tropical Degree of Cones in the Secondary Fan
نویسنده
چکیده
Abstract. Given a lattice polytope, the secondary fan is a combinatorial structure on the space of all tropical polynomials supported on the polytope. We consider rationally weighted formal sums of cones in the secondary fan as an approximation to a tropical variety. By using tropical intersection theory, we are able to intersect these formal sums with point-condition-imposing hyperplanes. Formal sums satisfying certain conditions give intersection numbers that are invariant of the choice of point-conditions. These intersection numbers are generalizations of classical enumerative invariants. By computing the degree of cones corresponding to nodal tropical curves, we are able to relate these invariants to Mikhalkin’s work on Gromov-Witten invariants.
منابع مشابه
Arithmetics and Combinatorics of Tropical Severi Varieties of Univariate Polynomials
We give a description of the tropical Severi variety of univariate polynomials of degree n having two double roots. We show that, as a set, it is given as the union of three explicit types of cones of maximal dimension n − 1, where only cones of two of these types are cones of the secondary fan of {0, . . . , n}. Through Kapranov’s theorem, this goal is achieved by a careful study of the possib...
متن کاملStar-convex functions on tropical linear spaces of complete graphs
Given a fan ∆ and a cone σ ∈ ∆, let star(σ) be the set of cones that contain σ and are one dimension bigger than σ. In this paper we study two cones of piecewise linear functions defined on ∆: the cone of functions which are convex on star(σ) for all cones, and the cone of functions which are convex on star(σ) for all cones of codimension 1. We give nice combinatorial descriptions for these two...
متن کاملAlgebraic Properties of Generic Tropical Varieties
We show that the algebraic invariants multiplicity and depth of a graded ideal in the polynomial ring are closely connected to the fan structure of its generic tropical variety in the constant coefficient case. Generically the multiplicity of the ideal is shown to correspond directly to a natural definition of multiplicity of cones of tropical varieties. Moreover, we can recover information on ...
متن کاملThe Birational Geometry of Tropical Compactifications
We study compactifications of subvarieties of algebraic tori using methods from the still developing subject of tropical geometry. Associated to each ``tropical" compactification is a polyhedral object called a tropical fan. Techniques developed by Hacking, Keel, and Tevelev relate the polyhedral geometry of the tropical variety to the algebraic geometry of the compactification. We compare thes...
متن کاملThe Effect of Pine Cones Aqueous Extract on Renal Function in Male Rats
Background:Antioxidant effects of cypress cones extract have been previously demonstrated. In this study, the protective effect of cypress cones extract was investigated. Gentamicin, an aminoglycoside antibiotic administrated for the treatment of gram-negative bacteria infections, was used for nephrotoxicity induction. Methods: In this study, 60 wistar male ra...
متن کامل