Tropical images of intersection points

@article{Morrison2014TropicalIO,
  title={Tropical images of intersection points},
  author={Ralph Morrison},
  journal={Collectanea Mathematica},
  year={2014},
  volume={66},
  pages={273-283}
}
  • Ralph Morrison
  • Published 3 March 2014
  • Mathematics
  • Collectanea Mathematica
A key issue in tropical geometry is the lifting of intersection points to a non-Archimedean field. Here, we ask: where can classical intersection points of planar curves tropicalize to? An answer should have two parts: first, identifying constraints on the images of classical intersections, and, second, showing that all tropical configurations satisfying these constraints can be achieved. This paper provides the first part: images of intersection points must be linearly equivalent to the stable… 

Lifting tropical self intersections

Tropical geometry for Nagata's conjecture and Legendrian curves

In this thesis, tropical methods in singularity theory and legendrian geometry are developed; tropical modifications are surveyed and several technical statements about them are proven. In more

A generalization of lifting non-proper tropical intersections

  • Xiang He
  • Mathematics
    Journal of Pure and Applied Algebra
  • 2019

Linear Tropicalizations

Let X be a closed algebraic subset of A(K) where K is an algebraically closed field complete with respect to a nontrivial nonArchimedean valuation. We show that there is a surjective continuous map

Tropical and non-Archimedean curves

Tropical geometry is young field of mathematics that connects algebraic geometry and combinatorics. It considers “combinatorial shadows” of classical algebraic objects, which preserve information

A guide to tropical modifications

This paper surveys tropical modifications, which have already become folklore in tropical geometry. Tropical modifications are used in tropical intersection theory and in a study of singularities.

Tropical Geometry

  • Ralph Morrison
  • Mathematics
    Foundations for Undergraduate Research in Mathematics
  • 2020
Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of

Lifting tropical bitangents

References

SHOWING 1-10 OF 13 REFERENCES

Lifting Tropical Intersections

We show that points in the intersection of the tropicaliza- tions of subvarieties of a torus lift to algebraic intersection points with expected multiplicities, provided that the tropicalizations

Lifting non-proper tropical intersections

We prove that if X, X' are closed subschemes of a torus T over a non-Archimedean field K, of complementary codimension and with finite intersection, then the stable tropical intersection along a

A guide to tropicalizations

Tropicalizations form a bridge between algebraic and convex geometry. We generalize basic results from tropical geometry which are well-known for special ground elds to arbitrary non-archimedean

Inflection Points of Real and Tropical Plane Curves

We prove that Viro's patchworking produces real algebraic curves with the maximal number of real inflection points. In particular this implies that maximally inflected real algebraic $M$-curves

Nonarchimedean geometry, tropicalization, and metrics on curves

We develop a number of general techniques for comparing analytifications and tropicalizations of algebraic varieties. Our basic results include a projection formula for tropical multiplicities and a

Linear systems on tropical curves

A tropical curve Γ is a metric graph with possibly unbounded edges, and tropical rational functions are continuous piecewise linear functions with integer slopes. We define the complete linear system

Tropical geometry and its applications

From a formal perspective tropical geometry can be viewed as a branch of geometry manipulating with certain piecewise-linear objects that take over the role of classical algebraic varieties. This

Tropical Convexity and Canonical Projections

Using a potential theory on metric graphs "Gamma", we introduce the notion of tropical convexity to the space "RDiv^d(Gamma)" of effective R-divisors of degree d on "Gamma" and show that a natural

A Riemann–Roch theorem in tropical geometry

Recently, Baker and Norine have proven a Riemann–Roch theorem for finite graphs. We extend their results to metric graphs and thus establish a Riemann–Roch theorem for divisors on (abstract) tropical