Splitting the nodes and branches of a proper nodal curve

Let X be a proper nodal curve over a field K. Its singular subscheme Sing(X/K) is finite étale over K, and its branch scheme E is finite étale of rank two over Sing(X/K) (AlgebraicGeometry.Scheme.finrank_branchSchemeToSingularLocus). After one finite Galois extension L / K, both become disjoint unions of rational points: the singular subscheme of the base change X_L is Spec (Fin n → L), a labelling of the n nodes, and its branch scheme is Spec (Fin 2 → Fin n → L) over it, two copies of the nodes. So every node of X_L is a rational point with two rational branches, labelled by Fin 2, and the pairing involution exchanging the two branches through each node is the swap of the two copies.

The splitting applies to every chosen base change of the curve: the singular subscheme and the branch scheme commute with the flat base change Spec L → Spec K (AlgebraicGeometry.Scheme.isPullback_singularLocus_subschemeι, AlgebraicGeometry.Scheme.exists_isPullback_branchSchemeToSingularLocus). Once nodes and branches are split over L, the branch scheme is the trivial double cover of the singular subscheme (AlgebraicGeometry.Scheme.Hom.exists_split_of_finrank_eq).

This labels nodes and branches only. Splitting the geometric components is not asserted here.

Main results

• AlgebraicGeometry.Scheme.exists_split_branchScheme: over one finite Galois extension of the base field, the nodes of a proper nodal curve split into rational points and its branch scheme into two copies of them.

References

• The Stacks Project, Section 53.19, Nodal curves, Tag 0C46.
• The Stacks Project, Tag 0C58 (families of nodal curves).
• The Stacks Project, Tag 03QR (schemes étale over a point).

theorem AlgebraicGeometry.Scheme.exists_split_branchScheme {K : Type u} [Field K] (X : Scheme) [X.Over (Spec ↧K)] [IsProper (X ↘ Spec ↧K)] (hnodal : TauCeti.AlgebraicGeometry.AtWorstNodalOfRelativeDimensionOne (X ↘ Spec ↧K)) : ∃ (L : Type u) (x : Field L) (x_1 : Algebra K L) (_ : Module.Finite K L) (_ : IsGalois K L) (n : ℕ), ∀ (X' : Scheme) (x_4 : X'.Over (Spec ↧L)) (g : X' ⟶ X) (H : CategoryTheory.IsPullback g (X' ↘ Spec ↧L) (X ↘ Spec ↧K) (Spec.algebraMap K L)), ∃ (e : (singularLocus L X').subscheme ≅ Spec ↧(Fin n → L)) (e' : branchScheme L X' ≅ Spec ↧(Fin 2 → Fin n → L)), CategoryTheory.CategoryStruct.comp e.hom (Spec.algebraMap L (Fin n → L)) = CategoryTheory.CategoryStruct.comp (singularLocus L X').subschemeι (X' ↘ Spec ↧L) ∧ CategoryTheory.CategoryStruct.comp e'.hom (Spec.algebraMap (Fin n → L) (Fin 2 → Fin n → L)) = CategoryTheory.CategoryStruct.comp (branchSchemeToSingularLocus L X') e.hom

The nodes and branches of a proper nodal curve split over one finite Galois extension. There are a finite Galois extension L / K and n : ℕ such that, for every base change X' of the curve to L, the singular subscheme of X' is Spec (Fin n → L) as a scheme over L, and the branch scheme of X' is Spec (Fin 2 → Fin n → L) as a scheme over the singular subscheme, through the diagonal inclusion of Fin n → L into Fin 2 → Fin n → L. Thus the nodes are n rational points, each with two rational branches indexed by Fin 2, and swapping the two indices is the pairing involution of the branches. This includes the smooth case n = 0.