A stratification of moduli of arbitrarily singular curves

Abstract

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 ${⨆}_{\Gamma }{\mathcal{E}}_{\Gamma }$ of ${\mathcal{E}}_{g,n}$ indexed by generalized dual graphs and prove that each stratum ${\mathcal{E}}_{\Gamma }$ 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.

69 pages, Example 2.20 corrected, notation reworked, index of notation added, many smaller corrections