A stratification of moduli of arbitrarily singular curves

Abstract

We construct a stratification $\bigsqcup_\Gamma \mathscr{E}_\Gamma$ of moduli of arbitrarily singular reduced curves indexed by generalized dual graphs and prove that each stratum is a fiber bundle over a finite quotient of a product of $\mathcal{M}_{g,n}$'s. The fibers are locally closed subschemes of products of Ishii's "territories," projective moduli schemes parametrizing subalgebras of a fixed algebra. The setting for our stratification is a new moduli stack $\mathscr{E}_{g,n}$ of "equinormalized curves" which is a minor modification of the moduli space of all reduced, connected curves. We prove algebraicity of substacks $\mathscr{E}^{\delta,\delta'}_{g,n}$ where invariants $\delta, \delta'$ are fixed, coarsely stratifying $\mathscr{E}_{g,n}$, then refine this to the desired stratification $\mathscr{E}_\Gamma$. A key technical ingredient is the introduction of the invariant $\delta'$ which allows us to ensure conductors commute with base change.Comment: Corrected definition 1.7, added section comparing territories with crimping spaces. 37 pages, 4 figures. Comments very welcome