Rank of divisors on graphs: an algebro-geometric analysis

The divisor theory for graphs is compared to the theory of linear series on curves through the correspondence associating a curve to its dual graph. An algebro-geometric interpretation of the combinatorial rank is proposed, and proved in some cases.Comment: Final version incorporating referee remarks. Dedicated to Joe Harris for his sixtiest birthday. 23 page

Geometry of the theta divisor of a compactified jacobian

We study the theta divisor of the compactified jacobian of a nodal, possibly reducible, curve. We compute its irreducible components and give it a geometric interpretation consistent with the classical Brill-Noether theory of smooth curves. Some applications on hyperelliptic stable curves are appended.Comment: 36 pages. Final version, to appear in JEM

Linear series on semistable curves

The dimension of spaces of global sections for line bundles on semistable curves parametrized by the compactified Picard scheme is studied. The theorem of Riemann is shown to hold. The theorem of Clifford is shown to hold in the following cases: the curve has two components; the curve is any semistable curve and the degree is either 0 or 2g-2; the curve is stable, free from separating nodes, and the degree is at most 4. These results are all shown to be sharp. Applications to the Clifford index, to the combinatorial description of hyperelliptic curves, and to plane quintics are given.Comment: Minor revisions. New numbering matches the journal version. 41 page

Algebraic and tropical curves: comparing their moduli spaces

We construct the moduli space for equivalence classes of n-pointed tropical curves of genus g, together with its compactification given by weighted tropical curves, and establish some of its basic topological properties. We compare it to the moduli spaces of smooth and stable algebraic curves, from the combinatorial, the topological, and the Teichm\"uller point of view. The paper is written in an expository style, and it generalizes some results contained in sections 4-6 of arXiv:1001.2815v3.Comment: Changes in numbering to match the published versio