Recently, Baker and Norine (Advances in Mathematics, 215(2): 2007) found new analogies between graphs and Riemann surfaces by developing a Riemann-Roch machinery on a finite graph G. In this paper, we develop a general Riemann-Roch Theory for sub-lattices of the root lattice An by following the work of Baker and Norine, and establish connections between the Riemann-Roch theory and the Voronoi diagrams of lattices under certain simplicial distance functions. In this way, we rediscover the work of Baker and Norine from a geometric point of view and generalise their results to other sub-lattices of An. In particular, we provide a geometric approach for the study of the Laplacian of graphs. We also discuss some problems on classification of lattices with a Riemann-Roch formula as well as some related algorithmic issues.Recently, Baker and Norine [2] proved a graph theoretic analogue of the classical Riemann-Roch theorem for curves in algebraic geometry. The proof is combinatorial and makes use of chip-firing games [5] and parking functions on graphs. Several papers later extended the results of Baker and Norine to tropical curves [15,18,21]. The question treated in this paper is to characterize those lattices which admit a Riemann-Roch theorem for the corresponding analogue of the rank-function defined by Baker and Norine.

The Riemann-Roch theorem on a graph G is related to Alexander duality in combinatorial commutive algebra. We study the lattice ideal given by chip firing on G and the initial ideal whose standard monomials are the G-parking functions. When G is a saturated graph, these ideals are generic and the Scarf complex is a minimal free resolution. Otherwise, syzygies are obtained by degeneration. We also develop a self-contained Riemann-Roch theory for artinian monomial ideals.

No abstract

Abstract. The G-parking function ideal M G of a directed multigraph G is a monomial ideal which encodes some of the combinatorial information of G. It is an initial ideal of the toppling ideal I G , a lattice ideal intimately related to the chip-firing game on a graph. Both ideals were first studied by Cori, Rossin, and Salvy. A minimal free resolution for M G was given by Postnikov and Shaprio in the case when G is saturated, i. e., whenever there is at least one edge (u, v) for every ordered pair of distinct vertices u and v. They also raised the problem of an explicit description of the minimal free resolution in the general case. In this paper, we give a minimal free resolution of M G for any undirected multigraph G, as well as for a family of related ideals including the toppling ideal I G . This settles a conjecture of Manjunath and Sturmfels, as well as a conjecture of Perkinson and Wilmes.

We construct immersions of trivalent abstract tropical curves in the Euclidean plane and embeddings of all abstract tropical curves in higher dimensional Euclidean space. Since not all curves have an embedding in the plane, we define the tropical crossing number of an abstract tropical curve to be the minimum number of self-intersections, counted with multiplicity, over all its immersions in the plane. We show that the tropical crossing number is at most quadratic in the number of edges and this bound is sharp. For curves of genus up to two, we systematically compute the crossing number. Finally, we use our immersed tropical curves to construct totally faithful nodal algebraic curves via lifting results of Mikhalkin and Shustin.1991 Mathematics Subject Classification. 14T05.