The divisor theories on finite graphs and metric graphs were introduced systematically as analogues to the divisor theory on algebraic curves, and all these theories are deeply connected to each other via tropical geometry and non-archimedean geometry. In particular, rational functions, divisors and linear systems on algebraic curves can be specialized to those on finite graphs and metric graphs. Important results and interesting problems, including a graph-theoretic Riemann-Roch theorem, tropical proofs of conventional Brill-Noether theorem and Gieseker-Petri theorem, limit linear series on metrized complexes, and relations among moduli spaces of algebraic curves, non-archimedean analytic curves, and metric graphs are discovered or under intense investigations. The content in this thesis is divided into three main subjects, all of which are based on my research and are essentially related to the divisor theory of linear systems on metric graphs and its application to tropical geometry and non-archimedean geometry. Chapter 1 gives an overview of the background and a general introduction of the main results. Chapter 2 is on the theory of rank-determining sets, which are subsets of a metric graph that can be used for the computation of the rank function. A general criterion is provided for rank-determining sets and certain specific examples of finite rank-determining sets are presented. Chapter 3 is on the subject of a tropical convexity theory on linear systems on metric graphs. In particular, the notion of general reduced divisors is introduced as the main tool used to study this tropical convexity theory. Chapter 4 is on the subject of smoothing of limit linear series of rank one on re_ned metrized complexes. A general criterion for smoothable limit linear series of rank 1 is presented and the relations between limit linear series of rank 1 and possible harmonic morphisms to genus 0 metrized complexes are studied.

Ziel dieser Arbeit ist die explizite Berechnung gewisser geometrischer Zykel in Modulräumen von Kurven. In den letzten Jahren wurden Divisoren auf $\Mbar_{g,n}$ ausgiebig untersucht. Durch die Berechnung von Klassen in Kodimension 1 konnten wichtige Ergebnisse in der birationalen Geometrie der Räume $\Mbar_{g,n}$ erzielt werden. In Kapitel 1 geben wir einen Überblick über dieses Thema. Im Gegensatz dazu sind Klassen in Kodimension 2 im Großen und Ganzen unerforscht. In Kapitel 2 betrachten wir den Ort, der im Modulraum der Kurven vom Geschlecht 2k durch die Kurven mit einem Büschel vom Grad k definiert wird. Da die Brill-Noether-Zahl hier -2 ist, hat ein solcher Ort die Kodimension 2. Mittels der Methode der Testflächen berechnen wir die Klasse seines Abschlusses im Modulraum der stabilen Kurven. Das Ziel von Kapitel 3 ist es, die Klasse des Abschlusses des effektiven Divisors in $\Mbar_{6,1}$ zu berechnen, der durch punktierte Kurven [C, p] gegeben ist, für die ein ebenes Modell vom Grad 6 existiert, bei dem p auf einen Doppelpunkt abgebildet wird. Wie Jensen gezeigt hat, erzeugt dieser Divisor einen extremalen Strahl im pseudoeffektiven Kegel von $\Mbar_{6,1}$. Ein allgemeines Ergebnis über gewisse Familien von Linearsystemen mit angepasster Brill-Noether-Zahl 0 oder -1 wird eingeführt, um die Berechnung zu vervollständigen. / The aim of this thesis is the explicit computation of certain geometric cycles in moduli spaces of curves. In recent years, divisors of $\Mbar_{g,n}$ have been extensively studied. Computing classes in codimension one has yielded important results on the birational geometry of the spaces $\Mbar_{g,n}$. We give an overview of the subject in Chapter 1. On the contrary, classes in codimension two are basically unexplored. In Chapter 2 we consider the locus in the moduli space of curves of genus 2k defined by curves with a pencil of degree k. Since the Brill-Noether number is equal to -2, such a locus has codimension two. Using the method of test surfaces, we compute the class of its closure in the moduli space of stable curves. The aim of Chapter 3 is to compute the class of the closure of the effective divisor in $\M_{6,1}$ given by pointed curves [C,p] with a sextic plane model mapping p to a double point. Such a divisor generates an extremal ray in the pseudoeffective cone of $\Mbar_{6,1}$ as shown by Jensen. A general result on some families of linear series with adjusted Brill-Noether number 0 or -1 is introduced to complete the computation.