A stratification of moduli of arbitrarily singular curves

We introduce a new moduli stack ${\mathcal{E}}_{g,n}$ of ``equinormalized curves," closely related to the moduli space of all reduced, connected algebraic curves. We construct a stratification ${⨆}_{\mathrm{\Gamma }}{\mathcal{E}}_{\mathrm{\Gamma }}$ of ${\mathcal{E}}_{g,n}$ indexed by generalized dual graphs and prove that each stratum ${\mathcal{E}}_{\mathrm{\Gamma }}$ is a fiber bundle over a finite quotient of a product of ${\text{moduli}}_{g,n}$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.