| This thesis is devoted to the application of combinatorial techniques to the study of algebraic curves and their moduli spaces. We consider the algebraic rank, a combinatorial invariant of graphs that reflects the ranks of line bundles on all the curves with a fixed dual graph. It satisfies a Riemann--Roch theorem, bounds the rank of line bundles in a one parameter degeneration, and is closely related to the tropical rank of divisors on graphs. The relation between the tropical and algebraic rank leads to a version of Clifford's theorem for nodal curves. We then produce families of examples where the algebraic and tropical ranks differ. We study the Brill--Noether rank of metric graphs and tropical curves, which serves as a tropical analog for the dimension of the Brill--Noether locus. It varies upper seimcontinuously in families of tropical curves, and satisfies a specialization lemma, relating it with the dimension of the Brill--Noether locus of an algebraic curve. Finally, we consider smooth tropical plane quartics, show that they are never hyperelliptic, and admit seven families of tropical bitangent lines. |
