Weierstrass Points and Torsion Points on Tropical Curves

Abstract

We investigate two constructions on metric graphs, using the framework of tropical geometry. On a metric circle, i.e. a genus 1 tropical curve, each of these constructions produces a set of n points which are evenly spaced around the circle.

In the first part, we study Weierstrass points for a divisor on a metric graph (i.e. tropical curve). On a smooth algebraic curve, these are points which have ''special'' tangency behavior with respect to a given projective embedding. The Weierstrass locus on a metric graph may fail to be a finite set; we define a stable Weierstrass locus which is always finite. The stable locus agrees with the ''naive'' Weierstrass locus for a generic divisor class. We then investigate the distribution of Weierstrass points for a high-degree divisor. We show that in high degree, the distribution of Weierstrass points converges to Zhang's canonical measure. This measure can be described by probabilities of weighted spanning trees, or alternatively by current flows in an electrical resistor network. This distribution result is a tropical analogue of a theorem of Neeman concerning Weierstrass points on a complex algebraic curve.

In the second part, we consider how a metric graph under the Abel--Jacobi embedding intersects torsion points of its Jacobian. The Manin--Mumford conjecture states that this intersection is finite for a smooth algebraic curve of genus 2 or more; this conjecture was proved by Raynaud. For a metric graph, this conjecture fails when the edge lengths are all rational numbers. However, we show that the Manin--Mumford conjecture does hold for metric graphs (of genus 2 or more) which are biconnected and have edge lengths which are ''sufficiently irrational'' in a precise sense. Under these assumptions we prove a bound on the size of the intersection which depends only on the genus, namely 3g-3. Next we consider higher-degree analogues of the Manin--Mumford conjecture, concerning the maps sending d-tuples of points to the Jacobian. This motivates the definition of the ''independent girth'' of a graph, which gives a strict upper bound for d such that the higher-degree Manin--Mumford property holds. For a metric graph with large genus g, the independent girth is bounded above by the logarithm of g.