Hilbert-Mumford criterion for nodal curves

TL;DR

This paper proves that slope stable polarized weighted pointed nodal curves are Chow asymptotic stable using Hilbert-Mumford criterion, extending previous stability results and answering a longstanding question in algebraic geometry.

Contribution

It generalizes prior stability results for polarized nodal and weighted pointed curves and confirms Chow asymptotic stability via Hilbert-Mumford criterion.

Findings

Proves Chow asymptotic stability for slope stable polarized weighted pointed nodal curves.

Extends stability results of Caporaso and Hassett to a broader class of curves.

Addresses a question raised by Mumford and Gieseker on stability criteria.

Abstract

We prove by Hilbert-Mumford criterion that a slope stable polarized weighted pointed nodal curve is Chow asymptotic stable. This generalizes the result of Caporaso on stability of polarized nodal curves, and of Hasset on weighted pointed stable curves polarized by the weighted dualizing sheaves. It also solved a question raised by Mumford and Gieseker to prove the Chow asymptotic stability of stable nodal curves by Hilbert-Mumford criterion.