Connectedness of the moduli space of all reduced curves

Using moduli of equinormalized curves and Ishii’s theory of territories, we prove that the moduli stack of all reduced n-pointed algebraic curves of fixed arithmetic genus is connected.

💡 Research Summary

The paper establishes that the moduli stack U_{g,n} of all reduced, connected, n‑pointed algebraic curves of fixed arithmetic genus g is connected, and that every geometric fiber of the natural morphism U_{g,n} → Spec ℤ is geometrically connected. The result is surprising because the stack is known to have many irreducible components and to be highly non‑separated, due to the presence of non‑smoothable singularities. The authors achieve the connectedness by constructing explicit deformation paths that keep the normalization of a curve fixed while varying its singularities.

The technical core begins with a description of reduced curve singularities in terms of subalgebras of a product of power‑series rings A = ∏_{i=1}^m k