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 be a connected finite graph of genus , where . Our central object of study will be the notion of a *break divisor* on . Recall that a **divisor** on a graph is an assignment of an integer to each vertex of . The divisor is called **effective** if for all ; in this case, we often visualize by placing “chips” at . And the **degree** of is the sum of over all vertices , i.e., the total number of chips. By analogy with algebraic geometry, we write divisors on as formal sums , i.e., as elements of the free abelian group on .

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

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