A stratification of moduli of arbitrarily singular curves

Abstract

We introduce a new moduli stack $\mathscr{E}_{g,n}$ of ``equinormalized curves," closely related to the moduli space of all reduced, connected algebraic curves. We construct a stratification $\bigsqcup_Γ\mathscr{E}_Γ$ of $\mathscr{E}_{g,n}$ indexed by generalized dual graphs and prove that each stratum $\mathscr{E}_Γ$ is a fiber bundle over a finite quotient of a product of $\moduli_{g,n}$s. The fibers are moduli schemes parametrizing subalgebras of a fixed algebra, and are in principle explicitly computable as locally closed subschemes of products of Grassmannians. We thus obtain a remarkably explicit geometric description of moduli of reduced curves with arbitrary singularities.

Figures (6)

• Figure 1: A sketch of a singular pointed curve and the combinatorial type of its normalization map. The singularities, from left to right, are an $A_5$ singularity $y^2 = x^6$ of genus 2, a node, a cusp meeting a smooth branch transversely, and a cusp.
• Figure 2: Intermediate spaces in the construction of $\mathscr{E}^{\delta,c}_{g,n}$. The diagonal arrow is the one that we will later stratify into fiber bundles indexed by combinatorial type.
• Figure 3: Spaces in the construction of the stratification by combinatorial type.
• Figure 4: Sketches of the territory $\mathrm{Ter}^2_{A_{\vec{c}}}$ together with sketches of the associated singular curves $C$. The singularity isomorphic to $k\llbracket t^3, t^4, t^5\rrbracket$ is labeled "345." Black points of the territory indicate singularities with conductance 4, i.e. points of $\mathrm{Ter}^{2,4}_{A_{\vec{c}}}$, while lighter grey points of the territory correspond to singularities with conductance 3, i.e. points of $\mathrm{Ter}^{2,3}_{A_{\vec{c}}}$.
• Figure 5: Combinatorial types in $\mathscr{E}^{2,4}_{2,0}$. We label edges with their conductance and vertices with their genus. We omit conductances equal to 1 and genera equal to 0.

Theorems & Definitions (155)

• Definition 1.1
• Definition 1.2
• Proposition 1.3: see Proposition \ref{['prop:equinormalized_to_ugn']}
• Example 1.4
• Example 1.5
• Theorem I: see Theorem \ref{['thm:algebraicity']} and Theorem \ref{['thm:stratification_by_gamma']}
• Theorem II: see Proposition \ref{['prop:stratification_by_delta_deltaprime']} and Theorem \ref{['thm:stratification_by_gamma']}
• Theorem III: see Theorem \ref{['thm:algebraicity']} and Theorem \ref{['thm:stratification_by_gamma']}
• Theorem IV: see Theorem \ref{['thm:e_gamma_fiber_bundle']}
• Definition 2.1