Tropical curves and applications

Erwan Brugallé

Ejercicios para entregar.
Bibliografía.
Ejemplos de patchworking.

Short course proposal

This is a proposition for a short course at 1st year  master student level. The aim of this course is to give a basic introduction to tropical geometry, with some applications to real algebraic geometry.

Tropical geometry is the algebraic geometry build over the tropical semi-ring (T, « + », « . »)\$ where

T=RU{-infinity},  « a+b »=max(a,b), and  « a.b »=a+b.

The ``tropical roots'' of  polynomials defined upon this algebra are piecewise linear objects, and turn out to be easier to study than their classical analogues (i.e. zero set of polynomials defined over a field).  The tropical semi-ring is linked to the classical semi-ring (R⁺,+,.) by the so-called Maslov dequantization (see [Vir01]) : we transport the semi-ring structure on R⁺ to T by the homeomorphism log_t, and make t goes to infinity. Thank to this  dequantization process, many properties of classical algebraic varieties are reflected by tropical varieties. Conversely, one of the main issue in tropical geometry is then to understand which properties of tropical varieties can be lifted to classical algebraic varieties.

Recent years have seen a tremendous development in tropical geometry that both established the field as an area of its own right and unveiled its deep connections to numerous branches of pure and applied mathematics. As an example of application of tropical geometry, one of the most important is certainly the use of tropical methods in enumerative geometry. These methods were initiated in the seminal paper of Mikhalkin [Mik05], and were a breakthrough in both complex and real enumerative geometries. There is no doubt that tropical techniques promise to remain extremely fruitful in the future.

In this introductory course, I will mainly focus on the study of tropical curves in the plane, and how to use them to construct real algebraic curves.

Program of the course :

·         Introduction to tropical semi-ring and tropical polynomials; Maslov dequantization

·         Tropical curves in ; Kapranov Theorem.

·         Some basic tropical intersection theory; Bezout Theorem; Bernstein Theorem.

·         Application to real algebraic geometry: combinatorial patchworking and construction of real algebraic curves with  controlled topology.

For easy reading introductions to tropical geometry, I refer to [Bru09], [BPS08] (both in french), [Bru10] (in portuguese), [RGST05, [Mik07],  and [Gat].

About real algebraic geometry and Hilbert's 16th problem, one can read the survey [Vir84] and the website [Vir].

References

[BPS08]  N. Berline, A. Plagne, and C. Sabbah, editors. Géométrie tropicale. Editions de l’Ecole   Polytechnique, Palaiseau, 2008. available at

[Bru10] E. Brugallé. Um pouco de geometria tropical. Matematica Universitaria, 46:27–40, 2010. Translation from french by E. Amorim and N. Puignau.

[Gat] A. Gathmann. Tropical algebraic geometry. math.AG/0601322.

[Mik05] G. Mikhalkin. Enumerative tropical algebraic geometry in . J. Amer. Math. Soc., 18(2):313–377,  2005.

[Mik07] G. Mikhalkin. What is. . . a tropical curve? Notices Amer. Math. Soc., 54(4):511–513, 2007.

[RGST05] J. Richter-Gebert, B. Sturmfels, and T. Theobald. First steps in tropical geometry. In Idempotent  mathematics and mathematical physics, volume 377 of Contemp. Math., pages 289–317. Amer.  Math. Soc., Providence, RI, 2005.

[Vir] O. Ya. Viro. Patchworking. http://www.pdmi.ras.ru/∼olegviro/patchworking.html.

[Vir84] O. Ya. Viro. Progress in the topology of real algebraic varieties over the last six years. Russian  Math. Surveys, 41:55–82, 1984.

[Vir01] O. Viro. Dequantization of real algebraic geometry on logarithmic paper. In European Congress of  Mathematics, Vol. I (Barcelona, 2000), volume 201 of Progr. Math., pages 135–146. Birkh ̈user,  Basel, 2001.