Amoebas of algebraic varieties and tropical geometry

Details Amoebas of algebraic varieties and tropical geometry Amoebas of algebraic varieties and tropical geometry

2004-02-29

arxiv.org/abs/math/0403015v1

Mathematics

Algebraic Geometry

40 pages, 15 figures, a survey for the volume "Different faces in Geometry"

Scientific paper

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.

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