Tropical geometry of curves with large theta characteristics

In this dissertation we study tropicalization curves which have a theta characteristic with large rank. This fits in the more general framework of studying the limit linear series on a curve which degenerates to a singular curve. We explore this when the singular curve is not of compact type. In particular we investigate the case when dual graph of the degenerate curve is a chain of g-loops. The fundamental object under consideration is a family of curves over a complete discrete valuation ring. In the first half of the dissertation we study geometry of such a family. In the third chapter we study metric graphs and divisors on them. This could be a thought of as the theory of limit linear series on a curve of non-compact type. In the fourth chapter we make this connection via tropicalization. We consider a family of curves with smooth generic fiber X η of genus g such that the dual graph of the special fiber is a chain of g loops. The main theorem we prove is that if X η has a theta characteristic of rank r then there are at least r linear relations on the edge lengths of the dual graph.