The enumerative geometry of rational and elliptic tropical curves and a Riemann-Roch theorem in tropical geometry [Elektronische Ressource] / Michael Kerber

The enumerative geometry of rational and elliptictropical curves and a Riemann-Roch theorem intropical geometryMichael KerberVom Fachbereich Mathematikder Technischen Universit¨at Kaiserslauternzur Verleihung des akademischen GradesDoktor der Naturwissenschaften(Doctor rerum naturalium, Dr. rer. nat.)genehmigte Dissertation1. Gutachter: Prof. Dr. Andreas Gathmann2.hter: Prof. Dr. Ilia ItenbergVollzug der Promotion: 12.12.2008D 386Abstract:The work is devoted to the study of tropical curves with emphasis on their enumerativegeometry. Major results include a conceptual proof of the fact that the number ofrational tropical plane curves interpolating an appropriate number of general points isindependent of the choice of points, the computation of intersection products of Psi-classes on the moduli space of rational tropical curves, a computation of the number oftropical elliptic plane curves of given degree and fixed tropical j-invariant as well as a analogue of the Riemann-Roch theorem for algebraic curves.Mathematics Subject Classification (MSC 2000):14N35 Gromov-Witten invariants, quantum cohomology51M20 Polyhedra and polytopes; regular figures, division of spaces14N10 Enumerative problems (combinatorial problems)Keywords:Tropical geometry, tropical curves, enumerative geometry, metric graphs....dedicated to my parents — in love and gratitudeContentsPreface iiiTropical geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .