A stratification of moduli of arbitrarily singular curves

Author:

Sebastian Bozlee, Christopher Guevara, David Smyth

Keyword:

Mathematics, Algebraic Geometry, Algebraic Geometry (math.AG)

journal:

date:

2023-07-18 16:00:00

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.

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