21 research outputs found
Combinatorics and degenerations in algebraic geometry: Hurwitz numbers, Mustafin varieties and tropical geometry
Degenerationstechniken sind mächtige Werkzeuge in der Untersuchung verschiedener Objekte der algebraischen Geometrie. Sie ermöglichen es geometrische Objekte mit Hilfe kombinatorischer Methoden zu untersuchen. Diesem großen Nutzen geht jedoch häufig die große Herausforderung voraus den Zusammenhang zur Kombinatorik zu konkretisieren. Das junge Gebiet der tropischen Geometrie bietet einen konzeptionellen Rahmen, um diesen Zusammenhang herzustellen. In dieser Arbeit untersuchen wir verschiedene Probleme der algebraischen Geometrie mit Hilfe degenerativer und kombinatorischer Methoden. Diese Probleme lassen sich wiederum in drei Themengebiete unterteilen: Enumeration von Überlagerungen (Hurwitz Zahlen), Degenerationen von projektiven Räumen (Mustafin varietäten), Modulräume tropischer Kurven und treue Tropikalisierungen
Triply mixed coverings of arbitrary base curves: Quasimodularity, quantum curves and a mysterious topological recursions
Simple Hurwitz numbers enumerate branched morphisms between Riemann surfaces
with fixed ramification data. In recent years, several variants of this notion
for genus base curves have appeared in the literature. Among them are
so-called monotone Hurwitz numbers, which are related to the HCIZ integral in
random matrix theory and strictly monotone Hurwitz numbers which count certain
Grothendieck dessins d'enfants. We generalise the notion of Hurwitz numbers to
interpolations between simple, monotone and strictly monotone Hurwitz numbers
to any genus and any number of arbitrary but fixed ramification profiles. This
yields generalisations of several results known for Hurwitz numbers. When the
target surface is of genus one, we show that the generating series of these
interpolated Hurwitz numbers are quasimodular forms. In the case that all
ramification is simple, we refine this result by writing this series as a sum
of quasimodular forms corresonding to tropical covers weighted by Gromov-Witten
invariants. Moreover, we derive a quantum curve for monotone and Grothendieck
dessins d'enfants Hurwitz numbers for arbitrary genera and one arbitrary but
fixed ramification profile. Thus, we obtain spectral curves via the
semiclassical limit as input data for the CEO topological recursion.
Astonishingly, we find that the CEO topological recursion for the genus
spectral curve of the strictly monotone Hurwitz numbers compute the monotone
Hurwitz numbers in genus . Thus, we give a new proof that monotone Hurwitz
numbers satisfy CEO topological recursion. This points to an unknown relation
between those enumerants. Finally, specializing to target surface
, we find recursions for monotone and Grothendieck dessins
d'enfants double Hurwitz numbers, which enables the computation of the
respective Hurwitz numbers for any genera with one arbitrary but fixed
ramification profile.Comment: 41 page