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.)

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.

Before formulating the combinatorial analogue of this result which Norine and I discovered, I want to briefly reminisce about how this result came about.  In the summer of 2006, my Georgia Tech REU (Research Experience for Undergraduates) student Dragos Ilas worked on a graph-theoretic conjecture which I had made some time earlier.  Dragos spent eight weeks working on the problem and compiled a lot of experimental evidence toward my conjecture.  He gave a talk about the problem one Friday toward the end of the summer in an REU Mini-Conference that I was organizing at Georgia Tech.  Serguei Norine (then a postdoc working with my colleague Robin Thomas) was in the audience.  On Monday morning, Serguei knocked on my office door and showed me an extremely clever proof of my conjecture.  I told Serguei about my real goal, which was to prove a graph-theoretic analogue of the Riemann-Roch theorem.  I outlined what I had in mind and within a week, we had exactly the kind of Riemann-Roch formula that I had hoped for… thanks in large part to Serguei’s amazing combinatorial mind! Continue reading

The Jacobian of the skeleton and the skeleton of the Jacobian

My new Georgia Tech colleague Joe Rabinoff and I recently posted this paper to the arXiv, entitled The skeleton of the Jacobian, the Jacobian of the skeleton, and lifting meromorphic functions from tropical to algebraic curves.  In this post I will attempt to give some context and background for that paper.

Suppose X is an algebraic variety over a complete non-Archimedean field K.  For simplicity of exposition, I will assume throughout that K is algebraically closed and that the value group \Lambda = {\rm val} (K^\times) is nontrivial.  The set of points X(K) possesses a natural (Hausdorff) analytic topology, but X(K) is totally disconnected and not even locally compact in this topology.  This is bad for all sorts of reasons.  Over the years there have been many attempts at “fixing” this problem, beginning with the Tate-Grothendieck theory of rigid analytic spaces.  One of the most successful and elegant solutions to the problem is Berkovich’s theory of non-Archimedean analytic spaces.  One can associate to X, in a natural and functorial way, an analytic space X^{\rm an} which contains X(K) as a dense subspace but has much nicer topological properties.  For example, X^{\rm an} is a locally compact Hausdorff space which is locally contractible, and is arcwise connected if X is connected.  As in the theory of complex analytic spaces, X^{\rm an} is compact if and only if X^{\rm an} is proper.  For much more information about Berkovich analytic spaces, see for example these course notes of mine.

One of the nice features of Berkovich’s theory (established in full generality only very recently, by Hrushovski and Loeser) is that the space X^{\rm an} always admits a deformation retraction onto a finite polyhedral complex \Sigma.  In general, there is no canonical choice for \Sigma.  However, in certain special cases there is, e.g. when X is a curve of positive genus or X is an abelian variety.  (More generally, the existence of canonical skeleta is closely tied in with the Minimal Model Program in birational algebraic geometry, see for example this recent paper of Mustata and Nicaise.)  For the rest of this post, I will focus on the special cases of curves and abelian varieties. Continue reading