__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**=**R**U{-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 **R²**; 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.

**Potential**** Readings**

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

http://www.math.polytechnique.fr/xups/vol08.html.

**[Bru09]** E. Brugallé. *Un peu de
géométrie tropicale*. Quadrature, (74):10–22, 2009. available at http://people.math.jussieu.fr/∼brugalle/articles/Quadrature/Quadrature.pdf

**[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 R² .* 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. D*equantization**
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.