A stratification of moduli of arbitrarily singular curves

This paper introduces a new moduli stack of "equinormalized curves" and establishes a stratification indexed by generalized dual graphs, where each stratum is described as a fiber bundle over a quotient of products of classical moduli spaces with explicitly computable fibers, thereby providing a detailed geometric characterization of the moduli of reduced curves with arbitrary singularities.

The Gist

Imagine you are an architect trying to catalog every possible shape a building can take.

In the world of algebraic geometry, these "buildings" are called curves. Some are smooth and perfect, like a polished marble ring. Others are messy, with kinks, sharp points, or places where the structure folds over itself (singularities).

For a long time, mathematicians had a great way to organize the perfect buildings (smooth curves) and the ones with simple kinks (nodes, like a figure-eight). They used a tool called a Dual Graph. Think of this as a simple stick-figure map: circles represent the smooth rooms of the building, and lines represent the doors connecting them. If you know the map, you know the basic layout.

The Problem:
What happens when the building has a really weird, complex mess? Maybe a sharp spike (a cusp), or a point where three walls meet at once, or a "ramphoid cusp" (a very sharp, bird-beak shape). The old stick-figure maps weren't detailed enough to describe these messy structures. Trying to catalog every possible messy curve was like trying to sort a pile of melted, tangled spaghetti by just looking at the plate—it seemed impossible because the tangles could be infinite and chaotic.

The Solution: The "Equinormalized" Blueprint
The authors of this paper (Bozlee, Guevara, and Smyth) came up with a brilliant new way to organize these messy curves. They didn't try to study the messy building directly. Instead, they studied the clean, underlying blueprint and the instructions for how to glue it back together.

Here is the analogy:

1. The Normalization (The Clean Blueprint):
   Imagine you have a crumpled, knotted piece of paper (the messy curve). If you carefully unfold it and smooth it out, you get a flat, perfect sheet of paper (the "normalization"). This sheet is easy to understand.

   • In the paper: This is the smooth curve $\tilde{C}$.

2. The Gluing Instructions (The Subalgebra):
   The messy curve is just the flat sheet with specific instructions on how to tape points together.

   • Example: "Take point A and point B and tape them together to make a hole."
   • Example: "Take point C and point D and tape them so they form a sharp spike."
   • In the paper: This is the "subalgebra" or the "territory." It's the mathematical rulebook that says exactly how the points are identified.

3. The New Map (Combinatorial Types):
   The authors realized that instead of just drawing a stick figure, they could draw a much more detailed map called a Combinatorial Type.

   • Circles: Still represent the smooth rooms.
   • Squares: Represent the messy "knots" or singularities.
   • Lines: Represent the "branches" of the knot connecting to the rooms.
   • Labels: They add numbers to tell you how the points are glued (e.g., "glue these two points with a simple tape" vs. "glue these three points with a complex knot").

The "Territory" Analogy
The paper introduces a concept called a Territory. Imagine you have a specific knot (a singularity). You want to find all the ways you can create that knot using your flat sheet.

• Some ways of gluing are very rigid (only one way to do it).
• Some ways allow you to wiggle the points a little bit before gluing them.
• The "Territory" is a special mathematical space that contains every single possible way to create that specific knot.

The authors proved that for any messy curve, you can break it down into:

1. A smooth base (the rooms).
2. A specific "Territory" (the instructions for the knots).

Why is this a big deal?
Before this paper, studying messy curves was like trying to navigate a foggy forest with no map. You knew the trees were there, but you couldn't see the paths.

Now, the authors have built a stratification. Think of this as a giant, multi-layered filing cabinet:

• Layer 1: The smooth curves (easy to file).
• Layer 2: Curves with simple knots (filed by the type of knot).
• Layer 3: Curves with complex knots (filed by the "Combinatorial Type" map).

They showed that each layer is a fiber bundle. In everyday language, this means the messy part of the problem is separated from the smooth part.

• The "Smooth Part" is like the floor of a room (well-understood).
• The "Messy Part" is like the furniture in the room.
• The paper proves that the furniture always sits on the floor in a predictable, organized way. You can study the floor and the furniture separately, and then put them together to understand the whole room.

The "Conductance" Metaphor
The paper also introduces a concept called Conductance. Imagine the "glue" used to tape the points together.

• Some glues are weak (simple nodes).
• Some glues are super strong and complex (cusps).
• The authors realized that if you group the curves by the "strength" of their glue, the math becomes much easier. It's like sorting your tools not just by type (hammer, screwdriver), but by size and power. This sorting trick allows them to prove that their new map is mathematically sound and doesn't fall apart.

The Bottom Line
This paper gives mathematicians a universal instruction manual for building and categorizing any kind of algebraic curve, no matter how messy or singular it is.

Instead of getting lost in the chaos of "tangled spaghetti," they now have a system to:

1. Unfold the spaghetti into a flat sheet.
2. Read the specific "gluing instructions" (the territory).
3. Organize the result into a neat, structured library (the stratification).

This opens the door to solving many other problems in geometry, like calculating the "volume" of these shapes or understanding how they change over time, because now they finally have a clear map of the territory.

Technical Summary

Here is a detailed technical summary of the paper "A Stratification of Moduli of Arbitrarily Singular Curves" by Sebastian Bozlee, Christopher Guevara, and David Smyth.

1. Problem Statement

The moduli stack of stable pointed curves, $\mathcal{M}_{g,n}$, is well-understood and possesses a stratification indexed by dual graphs, which encode the topological type of nodal curves. However, the moduli stack $\mathcal{U}_{g,n}$ of all connected, reduced, pointed curves with arbitrary isolated singularities (including cusps, tacnodes, and higher-order singularities) lacks a comparable, explicit geometric description.

Directly stratifying $\mathcal{U}_{g,n}$ is difficult because the deformation spaces of arbitrary singularities are complex and not well-behaved under base change. The authors aim to construct a stratification of a related moduli space that captures the combinatorial data of singular curves while providing a tractable geometric structure, ultimately allowing for the study of moduli of singular curves via the well-understood geometry of smooth curves and algebraic subalgebras.

2. Methodology and Strategy

The authors introduce a new moduli stack, $\mathcal{E}_{g,n}$, of "equinormalized curves." An object in this stack is a tuple $(\nu: \tilde{C} \to C, p_1, \dots, p_n)$, where:

• $\tilde{C}$ is a smooth, proper curve (the normalization).
• $C$ is a reduced, connected curve with arbitrary singularities.
• $\nu$ is the normalization map.
• $p_i$ are marked points on $\tilde{C}$ (lifting the markings on $C$).

The core strategy relies on the observation that a singular curve $C$ can be recovered from its normalization $\tilde{C}$ by specifying a subalgebra $\mathcal{O}_C \subset \nu_*\mathcal{O}_{\tilde{C}}$. The paper proceeds through the following steps:

A. Moduli of Subalgebras (Territories)

The authors generalize the concept of "territories" (originally introduced by Ishii) to families of algebras.

• Territories: They construct moduli schemes, called territories ($\text{Ter}^\delta_A$), parametrizing subalgebras of a fixed locally free algebra $A$ with a fixed corank $\delta$.
• Conductance Stratification: A major technical hurdle is that conductor ideals (which define the singularities) do not generally commute with base change. The authors resolve this by fixing the total conductance $c$ (the rank of the quotient of the normalization by the conductor). They prove that if the conductance is locally constant, the conductor ideal behaves well, allowing them to construct algebraic stacks $\mathcal{E}^{\delta, c}_{g,n}$ parametrizing equinormalized curves with fixed total delta invariant $\delta$ and total conductance $c$.

B. Combinatorial Types (Generalized Dual Graphs)

To stratify $\mathcal{E}^{\delta, c}_{g,n}$, the authors define combinatorial types $\Gamma$. These generalize dual graphs to arbitrary singularities:

• Vertices: Represent irreducible components of $\tilde{C}$ and singularities of $C$.
• Edges (Branches): Represent the preimages of singularities on $\tilde{C}$.
• Data: Includes genus of components, genus of singularities, incidence relations, and branch conductances (local invariants of the singularity).
• Unlike standard dual graphs, these types encode the specific "gluing" data required to reconstruct the singularity from the normalization.

C. Fiber Bundle Structure

The main geometric result is that each stratum $\mathcal{E}_\Gamma$ (curves of a fixed combinatorial type) is a fiber bundle over a base moduli space $\mathcal{M}_\Gamma$.

• Base ($\mathcal{M}_\Gamma$): A finite quotient of a product of moduli spaces of smooth curves $\mathcal{M}_{g,n}$. It parametrizes the normalization $\tilde{C}$ along with the unordered preimages of the singularities.
• Fiber: An explicitly computable moduli scheme parametrizing the subalgebras of a fixed algebra (the "territory" of the singularity).
• Topology: The map $\mathcal{E}_\Gamma \to \mathcal{M}_\Gamma$ is an étale fiber bundle (in characteristic 0).

3. Key Contributions and Results

Theoretical Framework

• Definition of $\mathcal{E}_{g,n}$: A rigorous construction of the stack of equinormalized curves, showing it is an algebraic stack (for fixed invariants) and that the map to $\mathcal{U}_{g,n}$ is finite-to-one.
• Algebraicity of Subalgebra Moduli: Proving that the moduli of subalgebras with fixed conductance are representable by schemes, overcoming the non-representability issues of general subalgebra moduli on non-reduced schemes.
• Stratification Theorems:
  • Theorem I & II: Existence of a stratification of $\mathcal{E}_{g,n}$ by combinatorial types $\Gamma$, where each stratum $\mathcal{E}_\Gamma$ is a locally closed substack.
  • Theorem III: Existence of a coarser stratification by numerical invariants $(\delta, c)$.
  • Theorem IV: The structural theorem stating that $\mathcal{E}_\Gamma \to \mathcal{M}_\Gamma$ is an étale fiber bundle with fiber equal to a territory of a fixed algebra.

Technical Innovations

• Conductance Fixing: The insight that fixing the rank of the conductor quotient ($c$) ensures the conductor ideal commutes with base change, which is crucial for the algebraicity of the moduli problem.
• Gluing Data and Territories: Decomposing the territory of a product algebra into a disjoint union of products of territories of singularities based on "gluing data" (how points are identified).
• Characteristic 0 Assumption: The stratification of the Hilbert scheme of points (used to fix branch conductances) requires characteristic 0 to ensure the quotient by symmetric groups behaves well (étale vs. fppf issues in positive characteristic).

Explicit Computations

The paper includes detailed computations for the case of genus 2 curves with total delta invariant 2 and conductance 4 ($\mathcal{E}^{2,4}_{2,0}$).

• They compute the specific territories for various singularity types (e.g., ramphoid cusps, tacnodes, ordinary triple points).
• They explicitly describe the 15 possible combinatorial types for this case and the closure relations between the strata (e.g., how a family of nodes can degenerate into a cusp).

4. Significance and Applications

• Explicit Geometry: The paper provides a remarkably explicit geometric description of moduli spaces of singular curves, reducing complex singular curve problems to the study of smooth curves and linear algebra (subalgebras of fixed algebras).
• Intersection Theory: The fiber bundle structure suggests that the Chow rings of these strata can be computed using the Chow rings of the base $\mathcal{M}_\Gamma$ and the fibers. This opens the door to calculating intersection numbers for moduli spaces of curves with worse-than-nodal singularities, which is relevant for the log minimal model program and compactifications of $\mathcal{M}_{g,n}$.
• Generalization of Dual Graphs: The introduction of "combinatorial types" extends the powerful tool of dual graphs (used for nodal curves) to the realm of arbitrary singularities, allowing for a systematic classification of degenerations.
• Connection to Crimping Spaces: The work clarifies the relationship between the authors' territories and "crimping spaces" (from van der Wyck's thesis), showing that crimping spaces are orbits within these territories under the action of automorphism groups.

5. Limitations and Future Work

• Characteristic 0: The current stratification relies on characteristic 0 for the behavior of the Hilbert scheme of points. The authors suspect the results hold in positive characteristic if the étale topology is replaced by the fppf topology, but this requires further technical work.
• Algebraicity of $\mathcal{E}_{g,n}$: While the strata $\mathcal{E}_\Gamma$ are shown to be algebraic stacks, the algebraicity of the full ambient stack $\mathcal{E}_{g,n}$ (without fixing invariants) is reserved for future work due to technical complexities.

In summary, this paper establishes a robust framework for studying moduli of singular curves by decomposing them into smooth components and algebraic gluing data, providing a pathway to compute their geometric invariants and understand their degenerations.