Tropical orbit spaces and moduli spaces of tropical curves [Elektronische Ressource] / Matthias Herold

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• Mathematics

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• Publié par: technische_universitat_kaiserslautern
• Publié le: 01 janvier 2011
• Nombre de lectures: 10
• Langue: English
• Poids de l'ouvrage: 1 Mo

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FACHBEREICH
MATHEMATIK
Kaiserslautern
&
Thesis INSTITUT DE
RECHERCHEpresented to receive the degree of Doktor der
´MATHEMATIQUENaturwissenschaften (Doctor rerum naturalium,
Dr.rer.nat) at Technische Universitat¨ Kaiserslautern, ´AVANCEE
resp. Doctorat de Mathematiques´ at l’Universite´ de
Strasbourg UMR 7501
specialty MATHEMATICS
Strasbourg
MatthiasHerold
Tropicalorbitspacesandmodulispacesof
tropicalcurves
D 386
Defended on January 25, 2011
in front of the jury
Mr. A. Gathmann, supervisor
Mr. I. Itenberg, supervisor
Mr. J-J. Risler, reviewer
Mr. E. Shustin, reviewer
www.mathematik.uni-kl.de
Mr. A. Oancea, scientific member
www-irma.u-strasbg.fr
Mr. G. Pfister, scientificA main result of this thesis is a conceptual proof of the fact that the weighted number of tropical
curves of given degree and genus, which pass through the right number of general points in the plane
r(resp., which pass through general points inR and represent a given point in the moduli space of genus
g curves) is independent of the choices of points. Another main result is a new correspondence theorem
between plane tropical cycles and plane elliptic algebraic curves.
Un principal résultat de la thèse est une preuve conceptionnelle du fait que le nombre pondéré de
courbes tropicales de degré et genre donnés qui passent par le bon nombre de points en position générale
2 rdansR (resp., qui passent par le bon nombre de points en position générale dansR et représentent un
point fixé dans l’espace de modules de courbes tropicales abstraites de genreg) ne dépend pas du choix de
points. Un autre principal résultat est un nouveau théorème de correspondance entre les cycles tropicaux
plans et les courbes algébriques elliptiques planes.
FACHBEREICH MATHEMATIK
Technische Universität Kaiserslautern
Postfach 3049
67653n
Germany
Tel: [phone] Fax: [phone]
[e-mail]
INSTITUT DE RECHERCHE MATHÉMATIQUE AVANCÉE
UMR 7501
Université de Strasbourg et CNRS
7 Rue René Descartes
67 084 STRASBOURG CEDEX
France
Tél. [phone] Fax [phone]
[e-mail]
Institut de Recherche
Mathématique Avancée
IRMA 2011/001
http://tel.archives-ouvertes.fr/tel-00550370/en/ISSN 0755-3390Contents
Preface v
Introduction en franc¸ais xiii
1 Polyhedral complexes 1
2 Moduli spaces 7
2.1 Moduli space of n-marked tropical curves . . . . . . . . . . . . . . . . . . . 7
2.2 Moduli space of parameterized labeled n-marked tropical curves . . . . . . . 9
3 Local orbit spaces 11
3.1 Tropical local orbit space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Morphisms of local orbit spaces . . . . . . . . . . . . . . . . . . . . . . . . 15
4 One-dimensional local orbit spaces 25
5 Moduli spaces for curves of arbitrary genus 29
5.1 Moduli spaces of abstract tropical curves . . . . . . . . . . . . . . . . . . . . 29
5.2 Moduli spaces of parameterized tropical curves . . . . . . . . . . . . . . . . 39
5.3 The number of curves is independent of the position of points . . . . . . . . . 42
6 Orbit spaces 47
6.1 Tropical orbit space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.2 Morphisms of orbit spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7 Moduli spaces of elliptic tropical curves 57
7.1 Moduli spaces of abstract tropical curves of genus 1 . . . . . . . . . . . . . . 57
7.2 Moduli spaces of parameterized tropical curves of genus 1 . . . . . . . . . . 62
7.3 Counting elliptic tropical curves with fixed j-invariant . . . . . . . . . . . . . 66
8 Correspondence theorems 69
8.1 Mikhalkin’s correspondence theorem . . . . . . . . . . . . . . . . . . . . . . 69
8.2 Correspondence theorem for elliptic curves with given j-invariant . . . . . . . 78
Bibliography 85
iiiPreface
ivPreface
Tropical geometry
Tropical geometry is a relatively new mathematical domain. The roots of tropical geometry
go back to the seventies (see [Be] and [BG]), but only ten years ago it became a subject on
its own. Tropical geometry has applications in several branches of mathematics such as enu-
merative geometry (e.g. [IKS], [M1]), symplectic geometry (e.g. [A]), number theory (e.g.
[G]) and combinatorics (e.g. [J]). A powerful tool in enumerative geometry are the so-called
correspondence theorems. These theorems establish an important correspondence between
complex algebraic curves satisfying certain constraints and tropical analogs of these curves.
One of the first results concerning correspondence theorems was achieved by G. Mikhalkin
(see [M1]). This theorem was proved again in slightly different form in [N], [NS], [Sh], [ST],
[T]. These results initiated the study of enumerative problems in tropical geometry (see for
example [GM1], [GM2], [GM3]). Dealing with counting problems, it is naturally to work
with moduli spaces. The first step in this direction was the construction of the moduli spaces
of rational curves given in [M2] and [GKM]. In [GKM] the authors developed some tools
to deal with enumerative problems for rational curves, using the notion of tropical fan. They
introduced morphisms between tropical fans and showed that, under certain conditions, the
weighted number of preimages of a point in the target of such a morphism does not depend
on the chosen point. After showing that the moduli spaces of rational tropical curves have
the structure of a tropical fan, they used this result to count rational curves passing through
given points.
Results
In the first part of this thesis we follow the approach of [GKM] and introduce similar tools for
enumerative problems concerning curves of positive genus. In the second part we establish a
new correspondence theorem. The main results of this thesis are as follows.
• We develop the definitions of (tropical) orbit spaces and (tropical) local orbit spaces
which are counterparts of a stack in algebraic geometry.
• We introduce morphisms between (tropical) orbit spaces and (tropical) local orbit
spaces.
vPreface
• For tropical (local) orbit spaces we show that the weighted number of preimages of a
point in the target of such a morphism does not depend on the chosen point.
• We equip the moduli spaces of tropical curves with the structure of a tropical local
orbit space.
• For the special case of moduli spaces of elliptic tropical curves we equip the moduli
spaces as well with the structure of a tropical orbit space.
• Using our results on tropical local orbit spaces, we give a more conceptual proof than
the authors of [KM] of the fact that the weighted number of plane tropical curves of
a given degree and genus which pass through the right number of points in general
2position inR is independent of the choice of a configuration of points.
• In the same way we prove that the weighted number of tropical curves of given degree
r rand genus inR which pass through the right number of points inR and which repre-
sent a fixed point in the moduli space of abstract genusg tropical curves is independent
of the choice of a configuration of points in general position.
• In the case of plane elliptic tropical curves of degreed we prove the independence of
the choice of a configuration of points and the choice of a type (which is thej-invariant
in this case) as well by using our results on tropical orbit spaces.
• We prove a correspondence between plane tropical cycles (of elliptic curves with big
j-invariant satisfying point constraints) and elliptic plane algebraic curves (satisfying
corresponding constraints).
The chapters 1 and 2 recall definitions and do not contain new results. The chapters 3, 4, 5, 6
and 7 are based on [H]. New results in chapter 8 are proposition 8.34, theorem 8.45 and the
conjecture 8.50.
Motivation
A relationship between tropical geometry and complex geometry was conjectured in 2000
by M. Kontsevich and was made precise by the so-called correspondence theorem by G.
Mikhalkin in [M1]. In the cases where such a connection is established, it suffices to count
tropical curves to get the number of corresponding algebraic objects. Therefore tropical
geometry became a powerful tool for enumerative geometry. In algebraic geometry one uses
moduli spaces in enumerative problems. Because of the mentioned relation, it would be
reasonable to construct moduli spaces in tropical geometry as well. For the construction of
moduli spaces in algebraic geometry one needs, in many cases, the notion of a stack. Put
simply, a stack is the quotient of a scheme by a group action. In this thesis we want to
make an attempt for the definition of a “tropical stack”. Since it is a first approach, we call
these objects tropical (local) orbit spaces (instead of calling them stacks). The definition of a
tropical orbit space avoids many technical problems. Therefore it is a useful definition to get
a first impression on the problems one wants to handle with a “tropical stack”. Nevertheless