Chow Stability of Curves of Genus 4 in P^3

TL;DR

This paper investigates the geometric invariant theory (GIT) construction of the moduli space of Chow semistable genus 4 curves in P^3, providing a classification and modular description of these curves.

Contribution

It offers a detailed classification of Chow stable and semistable genus 4 curves in P^3, including irreducible, nonreduced, and reducible cases, and describes the associated moduli space.

Findings

Classification of Chow stable and semistable curves of genus 4

Modular description of the moduli space of these curves

Insights into the birational map from stable to Chow semistable moduli

Abstract

In the paper, we study the GIT construction of the moduli space of Chow semistable curves of genus 4 in P^3. By using the GIT method developed by Mumford and a deformation theoretic argument, we give a modular description of this moduli space. We classify Chow stable or Chow semistable curves when they are irreducible or nonreduced. Then we work out the case when a curve has two components. Our classification provides some clues to understand the birational map from the moduli space of stable curves of genus 4 to the moduli space of Chow semistable curves of genus 4 in P^3.