The local model of a node as a relative curve

For a ring R and a ∈ R, the morphism Spec R[x, y] ⧸ (xy - a) ⟶ Spec R is the local model of a node in a family of curves: over a discrete valuation ring with uniformizer π and a = πⁿ with n > 0, its generic fibre is smooth and its special fibre is the union of two lines crossing transversally at the origin.

This file shows that this morphism is flat, locally of finite presentation, and of pure relative dimension one. These are the conditions, beside nodality of the geometric fibres, in the fibrewise characterization of families of curves with at worst nodal singularities. The fibre over a prime p of R is the spectrum of κ(p) ⊗[R] R[x, y] ⧸ (xy - a), that is, of the node algebra of the image of a in the residue field κ(p) (TauCeti.NodeAlgebra.baseChange), and over a field the node algebra is pure of dimension one (TauCeti.NodeAlgebra.isPureDimensional_primeSpectrum).

Main results

• TauCeti.NodeAlgebra.flat_spec: the local model of a node is flat.
• TauCeti.NodeAlgebra.locallyOfFinitePresentation_spec: it is locally of finite presentation.
• TauCeti.NodeAlgebra.fiberIso: its fibres are spectra of node algebras over residue fields.
• TauCeti.NodeAlgebra.pureRelativeDimension_spec: it has pure relative dimension one.

References

• Stacks Project, Example 55.14.1, Tag 0CDC, the local model xy = πⁿ of a node over a discrete valuation ring.

instance TauCeti.NodeAlgebra.flat_spec {R : Type u} [CommRing R] (a : R) : AlgebraicGeometry.Flat (AlgebraicGeometry.Spec.map (CommRingCat.ofHom (algebraMap R (NodeAlgebra R a))))

The local model of a node is flat.

instance TauCeti.NodeAlgebra.locallyOfFinitePresentation_spec {R : Type u} [CommRing R] (a : R) : AlgebraicGeometry.LocallyOfFinitePresentation (AlgebraicGeometry.Spec.map (CommRingCat.ofHom (algebraMap R (NodeAlgebra R a))))

The local model of a node is locally of finite presentation.

noncomputable def TauCeti.NodeAlgebra.fiberIso {R : Type u} [CommRing R] (a : R) (p : PrimeSpectrum R) : AlgebraicGeometry.Scheme.Hom.fiber (AlgebraicGeometry.Spec.map (CommRingCat.ofHom (algebraMap R (NodeAlgebra R a)))) p ≅ AlgebraicGeometry.Spec ↧(NodeAlgebra p.asIdeal.ResidueField ((algebraMap R p.asIdeal.ResidueField) a))

The fibre of the local model of a node over a prime p of R is the spectrum of the node algebra of the image of a in the residue field κ(p).

Equations

• One or more equations did not get rendered due to their size.

instance TauCeti.NodeAlgebra.pureRelativeDimension_spec {R : Type u} [CommRing R] (a : R) : AlgebraicGeometry.PureRelativeDimension 1 (AlgebraicGeometry.Spec.map (CommRingCat.ofHom (algebraMap R (NodeAlgebra R a))))

The local model of a node has pure relative dimension one: every fibre is a curve all of whose irreducible components are one-dimensional.