Source locality of nodal relative curves

Over an affine base, the singular subscheme of a relative curve pulls back to the singular subscheme of every étale neighbourhood, in particular of every open subscheme. Consequently, the scheme-theoretic nodal criterion is inherited by étale neighbourhoods, and it can be checked on an open cover of the source. This permits nodal affine charts to be glued, including the three charts of the blowup of a node.

Main results

• AffineAtWorstNodalOfRelativeDimensionOne.comp_of_etale: an étale neighbourhood of a nodal relative curve is nodal.
• affineAtWorstNodalOfRelativeDimensionOne_iff_of_openCover: nodality is local on the source for open covers.

References

• Stacks Project, Section 53.20, Tag 0C58.

theorem TauCeti.AlgebraicGeometry.AffineAtWorstNodalOfRelativeDimensionOne.comp_of_etale {R : Type u} [CommRing R] {X Y : AlgebraicGeometry.Scheme} {f : Y ⟶ AlgebraicGeometry.Spec ↧R} (h : AffineAtWorstNodalOfRelativeDimensionOne f) (g : X ⟶ Y) [AlgebraicGeometry.Etale g] : AffineAtWorstNodalOfRelativeDimensionOne (CategoryTheory.CategoryStruct.comp g f)

Pulling back along an étale morphism preserves the nodal-curve criterion: the singular subscheme of the étale source is the base change of the singular subscheme of the target. In particular this applies to restriction to an open subscheme.

theorem TauCeti.AlgebraicGeometry.AffineAtWorstNodalOfRelativeDimensionOne.of_openCover {R : Type u} [CommRing R] {Y : AlgebraicGeometry.Scheme} {f : Y ⟶ AlgebraicGeometry.Spec ↧R} (𝒰 : Y.OpenCover) (h : ∀ (j : 𝒰.I₀), AffineAtWorstNodalOfRelativeDimensionOne (CategoryTheory.CategoryStruct.comp (𝒰.f j) f)) : AffineAtWorstNodalOfRelativeDimensionOne f

The nodal-curve criterion over an affine base can be checked on an open cover of the source.

theorem TauCeti.AlgebraicGeometry.affineAtWorstNodalOfRelativeDimensionOne_iff_of_openCover {R : Type u} [CommRing R] {Y : AlgebraicGeometry.Scheme} (f : Y ⟶ AlgebraicGeometry.Spec ↧R) (𝒰 : Y.OpenCover) : AffineAtWorstNodalOfRelativeDimensionOne f ↔ ∀ (j : 𝒰.I₀), AffineAtWorstNodalOfRelativeDimensionOne (CategoryTheory.CategoryStruct.comp (𝒰.f j) f)

A relative curve over an affine base is nodal if and only if its restrictions to every member of an open cover are nodal.