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

The enumerationof planetropical curvesHannah MarkwigVom Fachbereich Mathematikder Technischen Universit at Kaiserslauternzur Verleihung des akademischen GradesDoktor der Naturwissenschaften(Doctor rerum naturalium, Dr. rer. nat.)genehmigte Dissertation.1. Gutachter: JProf. Dr. Andreas Gathmann2. Gutachter: Prof. Dr. Bernd SturmfelsVollzug der Promotion: 6. Juli 2006D 386Contents1. Introduction 11.1. Tropical geometry 11.2. Enumerative geometry 21.3. Tropical enumerative geometry 31.4. The content of this thesis 51.5. Acknowledgments 52. Motivation on plane tropical curves 72.1. The eld of Puiseux series 82.2. The tropical semiring 102.3. A combinatorial description of tropical curves 133. Classical concepts for the enumeration of plane curves 173.1. The moduli space of stable maps 18irr3.2. Kontsevich’s formula to determine N (d; 0) 24cplx3.3. The algorithm of Caporaso and Harris to determine N (d;g) 31cplx3.4. A short overview of curves on toric surfaces 424. The tropical enumerative problem in the plane 474.1. Abstract tropical curves 474.2. Parametrized tropical curves 514.3. The comparison of parametrized tropical curves with the de nitions of tropicalcurves from chapter 2 534.4. The moduli space of parametrized tropical curves 574.5. The tropical enumerative problem 664.6. The moduli space and the structure of polyhedral complexes 714.7. The proof of theorem 4.53 755.