September | 2017 | Matt Baker's Math Blog

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 →