Related papers: Logarithmic Picard groups, chip firing, and the co…

We introduce logarithmic Picard algebroids, a natural class of Lie algebroids adapted to a simple normal crossings divisor on a smooth projective variety. We show that such algebroids are classified by a subspace of the de Rham cohomology…

We study the algebraic rank of a divisor on a graph, an invariant defined using divisors on algebraic curves dual to the graph. We prove it satisfies the Riemann-Roch formula, a specialization property, and the Clifford inequality. We prove…

We determine the rational class and Picard groups of the moduli space of stable logarithmic maps in genus zero, with target projective space relative a hyperplane. For the class group we exhibit an explicit basis consisting of boundary…

We define the analogue of linear equivalence of graph divisors for the rotor-router model, and use it to prove polynomial time computability of some problems related to rotor-routing. Using the connection between linear equivalence for…

The paper by M. Baker and S. Norine in 2007 introduced a new parameter on configurations of graphs and gave a new result in the theory of graphs which has an algebraic geometry flavour. This result was called Riemann-Roch formula for graphs…

We propose a generalization of logarithmic and Schwarzenberger bundles over $\P^n=\P^n(\C)$ when the rank is greater than $n$. The first ones are associated to finite sets of points on $\P^{n\vee}$ and the second ones to curves with degree…

We construct the logarithmic and tropical Picard groups of a family of logarithmic curves and realize the latter as the quotient of the former by the algebraic Jacobian. We show that the logarithmic Jacobian is a proper family of…

A metrized complex of algebraic curves is a finite metric graph together with a collection of marked complete nonsingular algebraic curves, one for each vertex, the marked points being in bijection with incident edges. We establish a…

We investigate, using purely combinatorial methods, structural and algorithmic properties of linear equivalence classes of divisors on tropical curves. In particular, an elementary proof of the Riemann-Roch theorem for tropical curves,…

The logarithmic multiplicative group is a proper group object in logarithmic schemes, which morally compactifies the usual multiplicative group. We study the structure of the stacks of logarithmic maps from rational curves to this…

Baker and Norine introduced a graph-theoretic analogue of the Riemann-Roch theory. A central notion in this theory is the rank of a divisor. In this paper we prove that computing the rank of a divisor on a graph is NP-hard. The…

We define the logarithmic tautological rings of the moduli spaces of Deligne-Mumford stable curves (together with a set of additive generators lifting the decorated strata classes of the standard tautological rings). While these algebras…

We study the problem of computing the rank of a divisor on a finite graph, a quantity that arises in the Riemann-Roch theory on a finite graph developed by Baker and Norine (Advances of Mathematics, 215(2): 766-788, 2007). Our work consists…

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…

We investigate similarities between the category of vector spaces and that of polytopal algebras, containing the former as a full subcategory. In Section 2 we introduce the notion of a polytopal Picard group and show that it is trivial for…

Building on recent work of Robertson and Steger, we associate a C*-algebra to a combinatorial object which may be thought of as a higher rank graph. This C*-algebra is shown to be isomorphic to that of the associated path groupoid.…

We define the Laplacian matrix and the Jacobian group of a finite graph of groups. We prove analogues of the matrix tree theorem and the class number formula for the order of the Jacobian of a graph of groups. Given a group $G$ acting on a…

Let $k$ be a perfect field, and $X$ an irreducible smooth projective curve over $k$. We give a criterion for a vector bundle over $X$ to admit a logarithmic connection singular over a finite subset of $X$ with given residues, where residues…

In [5], Holroyd, Levine, M\'esz\'aros, Peres, Propp and Wilson characterize recurrent chip-and-rotor configurations for strongly connected digraphs. However, the number of steps needed to recur, and the number of orbits is left open for…

Given a connected reductive algebraic group G, we investigate the Picard group of the moduli stack of principal G-bundles over an arbitrary family of smooth curves.