Grigory Mikhalkin

Tropical Algebraic Geometry

Preface.- 1. Introduction to tropical geometry - Images under the logarithm - Amoebas - Tropical curves.- 2. Patchworking of algebraic varieties - Toric geometry - Viros patchworking method - Patchworking of singular algebraic surfaces - Tropicalization in the enumeration of nodal curves.- 3. Applications of tropical geometry to enumerative geometry - Tropical hypersurfaces - Correspondence theorem - Welschinger invariants.- Bibliography.

Real algebraic curves, the moment map and amoebas

In this paper we prove the topological uniqueness of maximal arrangements of a real plane algebraic curve with respect to three lines. More generally, we prove the topological uniqueness of a maximally arranged algebraic curve on a real toric surface. We use the moment map as a tool for studying the topology of real algebraic curves and their complexiflcations.

Amoebas of Algebraic Varieties and Tropical Geometry

This survey consists of two parts. Part 1 is devoted to amoebas. These are images of algebraic subvarieties in the complex torus under the logarithmic moment map. The amoebas have essentially piecewise-linear shape if viewed at large. Furthermore, they degenerate to certain piecewise-linear objects called tropical varieties whose behavior is governed by algebraic geometry over the so-called tropical semifield. Geometric aspects of tropical algebraic geometry are the content of Part 2. We pay special attention to tropical curves. Both parts also include relevant applications of the theories. Part 1 of this survey is a revised and updated version of an earlier prepreint of 2001.

Amoebas of Maximal Area

To any algebraic curve A in (*)2 one may associate a closed infinite region A in 2 called the amoeba of A. The amoebas of different curves of the same degree come in different shapes and sizes. All ...

Counting curves via lattice paths in polygons

Abstract This Note presents a formula for the enumerative invariants of arbitrary genus in toric surfaces. The formula computes the number of curves of a given genus through a collection of generic points in the surface. The answer is given in terms of certain lattice paths in the relevant Newton polygon. If the toric surface is P 2 or P 1 × P 1 then the invariants under consideration coincide with the Gromov–Witten invariants. The formula gives a new count even in these cases, where other computational techniques are available. To cite this article: G. Mikhalkin, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Tropical Open Hurwitz Numbers

We give a tropical interpretation of Hurwitz numbers extending the one discovered in \cite{CJM}. In addition we treat a generalization of Hurwitz numbers for surfaces with boundary which we call open Hurwitz numbers.

Genus 0 characteristic numbers of tropical projective plane

Finding the so-called characteristic numbers of the complex projective plane

Tropical geometry and its applications

{\mathbb C}P^2

Tropical Eigenwave and Intermediate Jacobians

is a classical problem of enumerative geometry posed by Zeuthen more than a century ago. For a given

Amoebas of Half-dimensional Varieties

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