# The Combinatorics of Break Divisors

I recently gave three lectures at Yale University for the Hahn Lectures in Mathematics.  The unifying theme of my talks was the notion of break divisor, a fascinating combinatorial concept related to the Riemann-Roch theorem for graphs.  Some applications of break divisors to algebraic geometry will be discussed in a follow-up post.

Break divisors on graphs

Let $G$ be a connected finite graph of genus $g = g(G)$, where $g := |E(G)| - |V(G)| + 1$.  Our central object of study will be the notion of a break divisor on $G$.  Recall that a divisor $D$ on a graph $G$ is an assignment of an integer $D(v)$ to each vertex $v$ of $G$.   The divisor $D$ is called effective if $D(v) \geq 0$ for all $v$; in this case, we often visualize $D$ by placing $D(v)$ “chips” at $v$.  And the degree of $D$ is the sum of $D(v)$ over all vertices $v$, i.e., the total number of chips.  By analogy with algebraic geometry, we write divisors on $G$ as formal sums $D = \sum_{v \in V(G)} D(v) (v)$, i.e., as elements of the free abelian group on $V(G)$.

A break divisor on $G$ is an effective divisor $D$ of degree $g$ such that for every connected subgraph $H$ of $G$, the degree of $D$ restricted to $H$ is at least $g(H)$.  In other words, there are $g(G)$ total chips and each connected subgraph $H$ contains at least genus-of-$H$ of these chips.  One surprising fact, proved in this paper (referred to henceforth as [ABKS]), is that the number of break divisors on $G$ is equal to the number of spanning trees of $G$. Continue reading

# Effective Chabauty

One of the deepest and most important results in number theory is the Mordell Conjecture, proved by Faltings (and independently by Vojta shortly thereafter). It asserts that if $X / {\mathbf Q}$ is an algebraic curve of genus at least 2, then the set $X({\mathbf Q})$ of rational points on $X$ is finite. At present, we do not know any effective algorithm (in theory or in practice) to compute the finite set $X({\mathbf Q})$. The techniques of Faltings and Vojta lead in principle to an upper bound for the number of rational points on $X$, but the bound obtained is far from sharp and is difficult to write down explicitly. In his influential paper Effective Chabauty, Robert Coleman combined his theory of p-adic integration with an old idea of Chabauty and showed that it led to a simple explicit upper bound for the size of $X({\mathbf Q})$ provided that the Mordell-Weil rank of the Jacobian of $X$ is not too large.  (For a memorial tribute to Coleman, who passed away on March 24, 2014, see this blog post.)

# Riemann-Roch theory for graph orientations

In this post, I’d like to sketch some of the interesting results contained in my Ph.D. student Spencer Backman’s new paper “Riemann-Roch theory for graph orientations”.

First, a bit of background.  In a 2007 paper, Emeric Gioan introduced the cycle-cocycle reversal system on a (finite, connected, unoriented) graph G, which is a certain natural equivalence relation on the set of orientations of G.  Recall that an orientation of G is the choice of a direction for each edge.  A cycle flip on an orientation ${\mathcal O}$ consists of reversing all the edges in a directed cycle in ${\mathcal O}$.  Similarly, a cocycle flip consists of reversing all the edges in a directed cocycle in ${\mathcal O}$, where a directed cocycle (also called a directed cut) is the collection of all oriented edges connecting a subset of vertices of G to its complement.  The cycle-cocycle reversal system is the equivalence relation on the set of orientations of G generated by all cycle and cocycle flips.  In his paper, Gioan proves (via a deletion-contraction recursion) the surprising fact that the number of equivalence classes equals the number of spanning trees in G.  A bijective proof of this result was subsequently obtained by Bernardi. Continue reading

# Reduced divisors and Riemann-Roch for Graphs

In an earlier post, I described a graph-theoretic analogue of the Riemann-Roch theorem and some of its applications.  In this post, I’d like to discuss a proof of that theorem which is a bit more streamlined than the one which Norine and I gave in our original paper [BN].  Like our original proof, the one we’ll give here is based on the concept of reduced divisors. Continue reading

# Riemann-Roch for Graphs and Applications

I plan to write several posts related to the Riemann-Roch Theorem for Graphs, which was published several years ago in this paper written jointly with Serguei Norine.  In this post I want to explain the statement of the theorem, give some anecdotal background, and mention a few applications which have been discovered in recent years.

The Riemann-Roch Theorem

The (classical) Riemann-Roch Theorem is a very useful result about analytic functions on compact one-dimensional complex manifolds (also known as Riemann surfaces).  Given a set of constraints on the orders of zeros and poles, the Riemann-Roch Theorem computes the dimension of the space of analytic functions satisfying those constraints.  More precisely, if $D$ denotes the set of constraints and $r(D)$ is the dimension of the space of analytic functions satisfying those constraints, then the Riemann-Roch theorem asserts that

$r(D) - r(K-D) = {\rm deg}(D) + 1 - g$

where $g$ is the genus (“number of holes”) of the Riemann surface $X$, ${\rm deg}(D)$ is the total number of constraints, and $K$ is the “canonical divisor” on $X$.  See the Wikipedia page for much more information.