The singular locus of a proper nodal curve

For a proper nodal curve over a field, the relative singular subscheme is finite étale over the field. Finiteness follows from properness and the fact that a nodal singular locus is unramified; flatness over a field then gives étaleness. This is the scheme that records the nodes before passing to a separable extension that labels them.

References

• Stacks Project, Tag 0C58

theorem AlgebraicGeometry.Scheme.isFinite_singularLocus_subschemeι_comp {R : Type u} [CommRing R] (X : Scheme) [X.Over (Spec ↧R)] [IsProper (X ↘ Spec ↧R)] (hnodal : TauCeti.AlgebraicGeometry.AtWorstNodalOfRelativeDimensionOne (X ↘ Spec ↧R)) : IsFinite (CategoryTheory.CategoryStruct.comp (singularLocus R X).subschemeι (X ↘ Spec ↧R))

The singular subscheme of a proper nodal relative curve over an affine base is finite. The nodal hypothesis is expressed by the scheme-theoretic criterion for an affine base.

theorem AlgebraicGeometry.Scheme.etale_singularLocus_subschemeι_comp {K : Type u} [Field K] (X : Scheme) [X.Over (Spec ↧K)] (hnodal : TauCeti.AlgebraicGeometry.AtWorstNodalOfRelativeDimensionOne (X ↘ Spec ↧K)) : Etale (CategoryTheory.CategoryStruct.comp (singularLocus K X).subschemeι (X ↘ Spec ↧K))

The singular subscheme of a nodal relative curve over a field is étale over that field. The nodal hypothesis is expressed by the scheme-theoretic criterion for an affine base.