Compact Moduli Spaces of Marked Cubic Plane Curves

Technical Summary: Compact Moduli Spaces of Marked Cubic Plane Curves

Problem Statement
This work addresses the construction and classification of compactifications for the moduli space of plane cubic curves marked by $n$ labeled points, considered up to projective equivalence. While compactifications of the moduli space of smooth curves of genus $g \geq 0$ with $n$ marked points (such as the Deligne-Mumford and Hassett compactifications) are well-understood via combinatorial structures like dual graphs and weights, the specific case of plane curves marked by points requires a different approach. The author focuses on the variation of Geometric Invariant Theory (VGIT) to construct these compactifications, specifically for degree $d=3$. The goal is to provide a complete description of the GIT walls and chambers within the ample cone of the parameter space, and to characterize the wall-crossing phenomena in terms of the singularities of the curves and the positions of the marked points.

Methodology
The paper employs Geometric Invariant Theory (GIT) to construct compactifications of the quasi-projective variety $C^{gen}_{n,d}/SL(3)$, where $C^{gen}_{n,d}$ represents the parameter space of degree $d$ plane curves with $n$ generic marked points. The methodology proceeds through the following steps:

1. Parameter Space and Linearization: The parameter space $C_{n,d}$ is defined as the subvariety of $\mathbb{P}(\Gamma(\mathbb{P}^2, \mathcal{O}(d))) \times (\mathbb{P}^2)^n$ cut out by incidence relations. The author constructs a family of $SL(3)$-linearized line bundles $L$ on this space, parameterized by a tuple of non-negative integers $(\gamma, w_1, \dots, w_n)$, where $\gamma$ acts as a weight on the curve and $w_i$ as weights on the marked points $p_i$.
2. Numerical Criterion: The stability of a marked curve is determined using the Hilbert-Mumford numerical criterion. The author reduces the criterion to normalized one-parameter subgroups of a maximal torus in $SL(3)$. This involves computing the numerical function $\mu_L(x, \lambda)$ based on the "support" of the curve and points under the action of the torus, utilizing the Mukai Order on monomials.
3. Ample Cone Determination: The paper first characterizes the $SL(3)$-ample cone $\Lambda(C_{n,d})$, identifying the region of weights $(\gamma, w_1, \dots, w_n)$ for which the locus of semi-stable marked curves is non-empty. This is achieved by analyzing the stability of sufficiently general (smooth) curves.
4. Wall Analysis for Cubics ($d=3$): For the specific case of cubic curves, the author identifies the "inner walls" of the ample cone. These walls correspond to line bundles where strictly semi-stable curves exist. By Lemma 4.1, these strictly semi-stable curves must possess a positive-dimensional $SL(3)$ stabilizer. The author classifies all reduced plane cubics with positive-dimensional stabilizers (Lemma 2.18) and determines the specific configurations of marked points that render them strictly semi-stable.
5. Wall-Crossing Description: The paper analyzes the transition of stability as one crosses these walls, identifying the generic stable curves on either side of a wall and their common degeneration (the strictly semi-stable curve on the wall).

Key Contributions and Results

• Characterization of the Ample Cone (Theorem 3.2): The paper provides an explicit description of the $SL(3)$-ample cone $\Lambda(C_{n,d})$ for any degree $d$. The cone is defined by a system of linear inequalities involving the total weight of points $W = \sum w_i$, individual weights $w_i$, and the curve weight $\gamma$. This generalizes previous results for $d=2$ (Giansiracusa and Simpson) and $n=1$ (Jensen).
• Classification of GIT Walls for Cubics (Theorem 5.1): For degree $d=3$, the paper identifies four distinct types of inner GIT walls, each associated with a specific degeneration of the cubic curve and a specific configuration of marked points:
  1. $3A_1$ (Three non-concurrent lines): Associated with points coinciding at a node.
  2. $A_2$ (Cuspidal curve): Associated with points coinciding at the cusp.
  3. $A_3$ (Conic and tangent line): Associated with points at the tacnode and on the linear component.
  4. $D_4$ (Cone over three points): Associated with points at the singularity and on the linear components.
     Theorem 5.1 provides the explicit hyperplane equations and boundary conditions for each wall type.
• Wall-Crossing Behavior (Theorem 5.2): The paper details the geometric changes occurring when crossing these walls. It describes the generic stable curves $S(T, I, \pm)$ on either side of a wall and their common degeneration $S(T, I, 0)$. For instance, crossing a wall associated with the $3A_1$ singularity transitions the moduli space from one containing irreducible nodal curves to one containing unions of a conic and a transverse line.
• Applications to Other Moduli Spaces:
  • Cubic Surfaces (Corollary 6.3): For $n=2$, the paper establishes an isomorphism between a specific GIT quotient of marked cubics and the moduli space of cubic surfaces containing a marked Eckardt point (which is isomorphic to the weighted projective space $\mathbb{P}(1, 2, 2, 3)$). This connects the work to Radu Laza's previous constructions.
  • Marked Elliptic Curves (Corollary 6.5): The paper constructs a space parametrizing plane cubics with $n$ points and an additional marked inflection point. It shows that the GIT quotient of this space compactifies the moduli space of marked elliptic curves $M_{1, n+1}$.

Significance and Claims
The paper claims to provide a "complete description" of the GIT walls and chambers for marked cubic plane curves, a task that is noted to be out of reach for arbitrary degrees due to the lack of a classification of degenerations for higher-degree curves. The work generalizes existing results for genus zero curves and specific low-degree cases.

The significance lies in the explicit combinatorial and geometric description of the boundary of these moduli spaces. By linking the GIT walls directly to the singularities of the curves (nodes, cusps, tacnodes, $D_4$ singularities) and the collision of marked points, the paper offers a concrete understanding of how the moduli space changes as the linearization varies. The author notes that the wall-crossing morphisms relate to weighted projective space bundles over the exceptional loci, providing a structural understanding of the birational geometry of these compactifications. The results serve as a foundation for understanding the birational map between $M_{1,n}$ and the constructed GIT quotients, which is identified as the focus of ongoing work.