Limit of Weierstrass Measure on Stable Curves

TL;DR

This paper investigates how Weierstrass measures behave on smooth curves as they degenerate to stable nodal curves, providing explicit descriptions at the boundary of the moduli space and analyzing related Bergman measures.

Contribution

It characterizes the limiting Weierstrass measures on stable curves in the Deligne-Mumford compactification, including boundary cases like rational curves.

Findings

Weierstrass measures on stable rational curves are explicitly determined.

Asymptotic behavior of Bergman measures is analyzed.

Limits of measures are described at the boundary of moduli space.

Abstract

The goal of the paper is to study the limiting behavior of the Weierstrass measures on a smooth curve of genus $g\geqslant 2$ as the curve approaches a certain nodal stable curve represented by a point in the Deligne-Mumford compactification $\bar{\mathcal M}_g$ of the moduli $\mathcal{M}_g$, including irreducible ones or those of compact type. As a consequence, the Weierstrass measures on a stable rational curve at the boundary of $\mathcal{M}_g$ are completely determined. In the process, the asymptotic behavior of the Bergman measure is also studied.