The Geometry of Break Divisors

I’d like to continue this discussion of break divisors on graphs & tropical curves by describing an interesting connection to algebraic geometry.  In this post, I’ll explain a beautiful connection to the theory of compactified Jacobians discovered by Tif Shen, a recent Ph.D. student of Sam Payne at Yale. Continue reading

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