The enumeration of plane tropical curves [Elektronische Ressource] / Hannah Markwig

Sujets

• Mathematics

Informations

• Publié par: technische_universitat_kaiserslautern
• Publié le: 01 janvier 2006
• Nombre de lectures: 21
• Langue: English
• Poids de l'ouvrage: 5 Mo

Extrait

The enumeration
of plane
tropical curves
Hannah Markwig
Vom Fachbereich Mathematik
der Technischen Universit at Kaiserslautern
zur Verleihung des akademischen Grades
Doktor der Naturwissenschaften
(Doctor rerum naturalium, Dr. rer. nat.)
genehmigte Dissertation.
1. Gutachter: JProf. Dr. Andreas Gathmann
2. Gutachter: Prof. Dr. Bernd Sturmfels
Vollzug der Promotion: 6. Juli 2006
D 386Contents
1. Introduction 1
1.1. Tropical geometry 1
1.2. Enumerative geometry 2
1.3. Tropical enumerative geometry 3
1.4. The content of this thesis 5
1.5. Acknowledgments 5
2. Motivation on plane tropical curves 7
2.1. The eld of Puiseux series 8
2.2. The tropical semiring 10
2.3. A combinatorial description of tropical curves 13
3. Classical concepts for the enumeration of plane curves 17
3.1. The moduli space of stable maps 18
irr3.2. Kontsevich’s formula to determine N (d; 0) 24
cplx
3.3. The algorithm of Caporaso and Harris to determine N (d;g) 31cplx
3.4. A short overview of curves on toric surfaces 42
4. The tropical enumerative problem in the plane 47
4.1. Abstract tropical curves 47
4.2. Parametrized tropical curves 51
4.3. The comparison of parametrized tropical curves with the de nitions of tropical
curves from chapter 2 53
4.4. The moduli space of parametrized tropical curves 57
4.5. The tropical enumerative problem 66
4.6. The moduli space and the structure of polyhedral complexes 71
4.7. The proof of theorem 4.53 75
5. The correspondence of tropical curves and lattice paths 90
5.1. Properties of parametrized tropical curves described in the dual language of
Newton subdivisions 91
5.2. The dual of a marked parametrized tropical curve 99
5.3. Lattice paths 1055.4. The correspondence of tropical curves and lattice paths 107
6. The correspondence of complex curves and tropical curves 114
6.1. J -holomorphic curves 115t
6.2. The \limit" of an amoeba | amoebas of J -holomorphic curves 117t
6.3. The number of complex curves whose limit is a given tropical curve 121
7. A tropical proof of Kontsevich’s formula 129
7.1. The enumerative problem for rational parametrized tropical curves 130
7.2. Tropical forgetful maps 132
7.3. Reducible curves and Kontsevich’s formula 138
8. The tropical Caporaso-Harris algorithm 150
8.1. Tropical curves that satisfy higher order tangency conditions to a line 151
8.2. The tropical Caporaso-Harris algorithm 153
8.3. The tropical Cap for irreducible curves 157
8.4. The correspondence of complex curves tangent (of higher order) to a line and
tropical curves with ends of higher weight 159
9. The Caporaso-Harris algorithm in the lattice path setting 165
9.1. Generalized lattice paths 165
9.2. The Caporaso-Harris algorithm for generalized lattice paths 167
9.3. The correspondence between tropical curves with ends of higher weight and
generalized lattice paths 173
References 175
Index 1771
1. Introduction
Tropical geometry is a rather new eld of algebraic geometry. The main idea is to replace
nalgebraic varieties by certain piece-wise linear objects in , which can be studied with the
aid of combinatorics. There is hope that many algebraically di cult operations become
easier in the tropical setting, as the structure of the objects seems to be simpler.
In particular, geometry shows promise for application in enumerative geometry.
Enumerative geometry deals with the counting of geometric objects that are determined
by certain incidence conditions. Until around 1990, not many enumerative questions had
been answered and there was not much prospect of solving more. But then Kontsevich
introduced the moduli space of stable maps which turned out to be a very useful concept
for the study of enumerative geometry. The idea of Kontsevich was motivated by physics,
more precisely, by string theory. Since then, enumerative geometry has gained a lot more
attention: not only from physicists, but also from mathematicians, as the theory of stable
maps has become rich and elaborated. However, a lot of questions remain open, and there
are still many mathematicians working in enumerative geometry.
Tropical geometry supplies many new ideas and concepts that could be helpful to answer
enumerative problems. However, as a rather new eld, tropical geometry has to be studied
more thoroughly. This thesis is concerned with the \translation" of well-known facts of
enumerative geometry to tropical geometry. We will rst give a short introduction to
tropical geometry and then explain the well-known results of enumerative geometry that
will be \translated" in this thesis.
1.1. Tropical geometry
Tropical geometry is so far best developed for plane curves. An idea Kontsevich proposed
and Mikhalkin elaborated in [23] is to apply the map
? 2 2Log : ( ) ! : (z;w)7! (logjzj; logjwj)
to a complex curve in a toric surface. The observation is that the image of a complex
2curve under this map looks roughly like a graph in with linear edges. When we shrink
the image to a certain limit, we end up with such a graph ful lling a condition called
the \balancing condition". Such a graph will be referred to as a tropical curve. An
analogous \deformation" of the complex numbers yields a semiring ( [f 1g ; max; +)
with operations max as addition and + as multiplication. This semiring has been known
to computer scientists before and is referred to as \tropical semiring" in honour of the
Brazilian mathematician and computer scientist Imre Simon (see for example [29]).
2As already mentioned, tropical curves look, shortly described, like graphs in which
ful ll certain conditions. The balancing condition allows to associate a dual to a tropical
2curve, which is a regular subdivision of a lattice polygon in (see section 2.3). Therefore,
the data of a tropical curve can be described purely combinatorially using lattice polygons
(respectively, their dual graphs).
A lot of work has been done to \translate" classical concepts to this tropical setting,
especially in enumerative geometry. A well-known problem of enumerative geometry is to
determine the numbers N (d;g) of complex genus g plane curves of degree d passingcplx
RRCRRRZ2
through 3d+g 1 points in general position. In [23], Mikhalkin associates a multiplicity to
each tropical curveC, which coincides with the number of complex curves that project to
C (under Log and taking the limit). He shows that the number N (d;g) is equal to thecplx
numberN (d;g) of tropical curves through 3d +g 1 points, counted with multiplicity.trop
This important result is referred to as Correspondence Theorem. We will describe it
more precisely in chapter 6. Furthermore, he computes N (d;g) = N (d;g) purelycplx trop
combinatorially using certain lattice paths in the lattice polygon dual to the tropical curves
(see chapter 5). Siebert and Nishinou extended the Correspondence Theorem to rational
curves in ann-dimensional toric variety [24]. Shustin showed the same for certain singular
plane curves [26].
But not only enumerative geometry has been translated to the tropical world: Izhakian
found an analogue to the duality of curves [16], Vigeland established a group law on
tropical elliptic curves [33] and Tabera dealt with a tropical Pappus’ Theorem [30], just
to mention a few.
Tropical research is not restricted to the translation of classically well-known facts, there
are actually new results shown by means of tropical geometry that have not been known
before. For example, Mikhalkin gave a tropical algorithm to compute the Welschinger
invariant for real curves [23] and Itenberg, Shustin and Kharlamov were able to estimate
the Welschinger invariant (even for large degrees where the computation using Mikhalkin’s
algorithm is too complicated) using tropical curves [15]. Furthermore, there are ideas by
Mikhalkin to compute Zeuthen numbers, that is the numbers of plane curves which do not
only satisfy the condition to pass through certain points, but also tangency conditions to
lines.
This shows that tropical geometry can indeed be a tool for a better understanding of
classical geometry.
1.2. Enumerative geometry
As already mentioned, enumerative geomtry deals with the counting of geometric objects
that satisfy given incidence conditions. The conditions must be chosen in such a way that
there is actually a nite number of objects that satisfy them.
The main strategy to count objects is to construct a moduli space which parametrizes
these objects. The special objects that satisfy one of the given incidence conditions will
then correspond to a subspace of the moduli space. In order to count objects that satisfy
all conditions, we have to intersect the subspaces corresponding to each condition, and
determine the number of points in the 0-dimensional intersection product. (The way the
incidence conditions were chosen - such that there are only nitely many objects satisfying
them - guarantees that this intersection is indeed 0-dimensional.) The moduli space of
stable maps is a moduli space that seems appropriate for a large class of enumerative
questions. Therefore, enumerative geometry deals basically with intersection theory on
this moduli space. The numbers which occur as intersection numbers on the moduli space
of stable maps are called Gromov-Witten invaria