I'm interested in learning tropical geometry. But my background in algebraic geometry is limited. I know basic facts about varieties in affine and projective space, but nothing about sheaves, schemes etc.

I wanted to know how much algebraic geometry does one need to understand the research literature in tropical geometry.

What are the other subjects which I must know before starting tropical geometry?

  • 7
    $\begingroup$ Well, according to Maclagan & Sturmfels in their draft of Introduction to Tropical Geometry, you need about the level of Ideals, Varieties, and Algorithms by Cox, Little, and O’Shea as background for the first five chapters (out of nine) of their book. (I've made this a comment since I do not address the research literature in tropical geometry, and in any case, I am paraphrasing Maclagan & Sturmfels rather than providing my own answer.) $\endgroup$
    – J W
    Dec 31 '11 at 7:57
  • 5
    $\begingroup$ You can learn already a great deal of tropical geometry with the knowledge you have. It is a very "visual" and "constructive" area of mathematics, and mainly works with hands-on definitions. Of course, there are some applications of tropical geometry to Grothendieck-style algebraic geometry, but if you do not know schemes and sheaves, you very probably will not be interested in these applications in the first place. So my suggestion is just: start reading some introductory paper right now, e.g. arxiv.org/abs/math/0601322 $\endgroup$ Dec 31 '11 at 15:29

Several years ago, I participated in a learning seminar in tropical algebraic geometry and collected several helpful survey articles. (This was before Maclagan and Sturmfels' book was written, which I suspect is excellent.)

Anyway, here were some of the most helpful intro points for me: Tropical Mathematics, First Steps in Tropical Geometry, Tropical Algebraic Geometry, Introduction to Tropical Geometry, The Tropical Grassmannian, The Number of Tropical Plane Curves Through Points in General Position.

Sturmfels, Speyer, and Gathmann all write very well, and Gathmann especially devotes considerable space to giving motivation for the field. Mikhalkin, of course, was the one who pioneered the idea of attacking challenging classical problems (such as counting the number of plane curves of genus $g$ and degree $d$ passing through $3d + g - 1$ points, which had just been solved by Capraso-Harris in the late 90s) using the tropical semifield.


I believe that only basic algebraic geometry is needed.

This on-line course on Tropical Geometry given by Bernd Sturmfels is based on this book, written by D. Maclagan and B. Sturmfels, and in the introduction they claim the following:

We have attempted to make the first part of the book (Chapters 1–5) accessible to readers with a minimal background in algebraic geometry, say, at the level of the undergraduate text book Ideals, Varieties, and Algorithms by Cox, Little, and O’Shea.

  • 1
    $\begingroup$ Thank you for the link to the on-line course. While my comment to the question mentions Maclagan & Sturmfels' book, I was unaware of the video lectures. $\endgroup$
    – J W
    Dec 31 '11 at 19:04

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