Algebraic rank and linear series on semistable curves

I will discuss results concerning the interplay between the geometry and combinatorics of line bundles on nodal curves. I will introduce the algebraic rank, a combinatorial invariant on graphs, that reflects the ranks of line bundles on all the curves in a stratum of span style=text-decoration: overlineiM/i/spansubig/i/sub. The invariant gives an upper bound on the rank of line bundles in a one parameter degeneration, and is closely related to the Baker-Norine rank of divisors on graphs. As an application, we obtain a version of Clifford’s theorem for nodal curves. Most of the results in the talk are based on joint work with Lucia Caporaso and Margarida Melo.