Limit of Weierstrass Measure on Stable Curves

Abstract

The goal of the paper is to study the limiting behavior of the Weierstrass measures on a smooth curve of genus $g⩾2$ 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.