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Compact Moduli Spaces of Marked Cubic Plane Curves

Aaron Goodwin

TL;DR

The paper develops a comprehensive VGIT framework for moduli of plane cubic curves with marked points, classifying all GIT walls for degree three and describing how wall-crossing corresponds to degenerations of cubics and the collision or alignment of marked points. It provides an explicit ample-cone description $\Lambda(\mathcal{C}_{n,3})$ and derives the four inner-wall types $W(3A_1,I)$, $W(A_2,I)$, $W(A_3,I,J)$, and $W(D_4,I)$ with concrete stability–instability criteria. The work connects marked cubics to other moduli spaces, giving explicit isomorphisms to $M^{(1,3)lab}_{pairs}(\tfrac{3}{2}-\epsilon) \cong \mathbb{P}(1,2,2,3)$ and showing $M_{1,n+1}$ arises as a VGIT quotient of a marked-cubic parameter space, via a dense open embedding and a Zariski–Main-type argument. Overall, it extends prior genus-zero results to cubics, clarifies how singularities and point configurations govern compactifications, and reveals concrete birational correspondences to classical moduli spaces of elliptic curves and cubic surfaces.

Abstract

We study compactifications of the moduli space of a plane cubic curve marked by \(n\) labeled points up to projective equivalence via Geometric Invariant Theory (GIT). Specifically, we provide a complete description of the GIT walls and show that the moduli-theoretic wall-crossing can be understood through analysis of the singularities of the plane curves and the position of the points.

Compact Moduli Spaces of Marked Cubic Plane Curves

TL;DR

The paper develops a comprehensive VGIT framework for moduli of plane cubic curves with marked points, classifying all GIT walls for degree three and describing how wall-crossing corresponds to degenerations of cubics and the collision or alignment of marked points. It provides an explicit ample-cone description and derives the four inner-wall types , , , and with concrete stability–instability criteria. The work connects marked cubics to other moduli spaces, giving explicit isomorphisms to and showing arises as a VGIT quotient of a marked-cubic parameter space, via a dense open embedding and a Zariski–Main-type argument. Overall, it extends prior genus-zero results to cubics, clarifies how singularities and point configurations govern compactifications, and reveals concrete birational correspondences to classical moduli spaces of elliptic curves and cubic surfaces.

Abstract

We study compactifications of the moduli space of a plane cubic curve marked by labeled points up to projective equivalence via Geometric Invariant Theory (GIT). Specifically, we provide a complete description of the GIT walls and show that the moduli-theoretic wall-crossing can be understood through analysis of the singularities of the plane curves and the position of the points.
Paper Structure (12 sections, 28 theorems, 28 equations, 3 figures, 1 table)

This paper contains 12 sections, 28 theorems, 28 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometric Invariant Theory
  4. The Numerical Criterion for marked plane curves
  5. Computing the stability of a marked curve
  6. Preliminary results on the geometry of plane cubics
  7. The wall and chamber decomposition
  8. Finding the inner walls
  9. Proof of the main theorems
  10. Applications to other moduli spaces
  11. Cubics with two points and cubic surfaces
  12. Plane Cubics and M1,n

Key Result

Theorem 2.3

For a given $G-$linearized line bundle $L$ on $X$, the semi-stable locus $X^{ss}$ is an open subset of $X$. The stable locus $X^s$ is an open subset of $X^{ss}$.

Figures (3)

  • Figure 1: (Left) The linearization polytope $\Delta(\mathcal{C}_{2,3})$ as in Section \ref{['sec:2pts']}. (Right) The locus of line bundles at which marked curves $(C,p_1,p_2)$ with the following pathologies are stable: A) $p_2$ supported at a node. B) $p_2$ supported at a cusp. C) $C$ is cuspidal. D) $p_1$ is supported on a linear component of $C$. E) $p_1$ and $p_2$ are both supported at inflection points. F) $p_1$ and $p_2$ collide.
  • Figure 2: Illustration of the wall crossing behavior found in Theorem \ref{['thm:wallcrossing']}. The curves $S(T,I, \pm)$ are stable in the chamber containing $L^{\pm}$ and unstable in the chamber containing $L^{\mp}$.
  • Figure 3: Singular cubics are determined up to projective equivalence by their singularities

Theorems & Definitions (63)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • Theorem 2.6
  • proof
  • Definition 2.7
  • ...and 53 more