A stratification of moduli of arbitrarily singular curves

TL;DR

This paper introduces a new moduli stack for equinormalized curves, providing an explicit geometric stratification related to dual graphs, and describes fibers as moduli schemes of subalgebras, advancing the understanding of singular algebraic curves.

Contribution

It constructs a stratification of the moduli stack of equinormalized curves indexed by dual graphs, with explicit fiber descriptions as moduli schemes of subalgebras, offering detailed geometric insights.

Findings

Stratification of the moduli stack by dual graphs.

Explicit description of fibers as moduli schemes of subalgebras.

Provides a geometric framework for singular curves with arbitrary singularities.

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