The distribution of Weierstrass points on a tropical curve

Abstract

We show that on a metric graph of genus $g$, a divisor of degree $n$ generically has $g(n-g+1)$ Weierstrass points. For a sequence of generic divisors on a metric graph whose degrees grow to infinity, we show that the associated Weierstrass points become distributed according to the Zhang canonical measure. In other words, the limiting distribution is determined by effective resistances on the metric graph. This distribution result has an analogue for complex algebraic curves, due to Neeman, and for curves over non-Archimedean fields, due to Amini.

Figures (12)

• Figure 1: Chip firing across a simple cut.
• Figure 2: Linear interpolation showing the divisor ${f}^{-1}_{\mathop{\mathrm{\Delta}}\nolimits}(\lambda)$.
• Figure 3: Break divisors (left) and non-break divisors (right).
• Figure 4: Metric graph $\Gamma$, on left, and ABKS decomposition of $\mathop{\mathrm{Pic}}\nolimits^2(\Gamma)$, on right.
• Figure 5: Edge length and deleted effective resistance on a metric graph.

Theorems & Definitions (107)

• Theorem A
• Theorem B
• Theorem C
• Theorem 1.1: Neeman N
• Theorem 1.2: Amini A-weier
• Remark 2.1: Linear equivalence as chip firing
• Definition 2.2
• Definition 2.3
• Theorem 2.4: Abel--Jacobi for metric graphs
• proof