Nodal and prestable models are detected on two fibres

Let R be a discrete valuation ring with fraction field K and residue field k. Every point of Spec R is its generic or its closed point, so a flat morphism X ⟶ Spec R locally of finite presentation is at worst nodal of relative dimension one exactly when its generic fibre X_K ⟶ Spec K and its special fibre X_k ⟶ Spec k are (TauCeti.AlgebraicGeometry.atWorstNodalOfRelativeDimensionOne_iff_genericFiber_and_specialFiber), and similarly for prestable families (TauCeti.AlgebraicGeometry.prestableFamily_iff_genericFiber_and_specialFiber).

For a model M of a curve C ⟶ Spec K over R, flatness and finite presentation are part of the data and the generic fibre is identified with C. Hence M is a nodal model exactly when C is a nodal curve and the special fibre of M is a nodal curve over k, and M is a prestable model exactly when it is proper and C and the special fibre are prestable curves.

Main results

• TauCeti.Model.atWorstNodalOfRelativeDimensionOne_toBase_iff: a model is at worst nodal exactly when its generic curve and its special fibre are.
• TauCeti.Model.prestableFamily_toBase_iff: a model is prestable exactly when it is proper and its generic curve and its special fibre are prestable.

References

• The Stacks Project, Section 55.18, Semistable reduction
• The Stacks Project, Lemma 53.20.1, Tag 0C58

theorem TauCeti.Model.atWorstNodalOfRelativeDimensionOne_toBase_iff {R K : Type u} [CommRing R] [IsDomain R] [IsDiscreteValuationRing R] [Field K] [Algebra R K] [IsFractionRing R K] {C : AlgebraicGeometry.Scheme} {toK : C ⟶ AlgebraicGeometry.Spec ↧K} (M : Model R K C toK) : AlgebraicGeometry.AtWorstNodalOfRelativeDimensionOne M.toBase ↔ AlgebraicGeometry.AtWorstNodalOfRelativeDimensionOne toK ∧ AlgebraicGeometry.AtWorstNodalOfRelativeDimensionOne (specialFiber R M.toBase).hom

A model over a discrete valuation ring is at worst nodal of relative dimension one exactly when its generic curve and its special fibre are at worst nodal.

theorem TauCeti.Model.prestableFamily_toBase_iff {R K : Type u} [CommRing R] [IsDomain R] [IsDiscreteValuationRing R] [Field K] [Algebra R K] [IsFractionRing R K] {C : AlgebraicGeometry.Scheme} {toK : C ⟶ AlgebraicGeometry.Spec ↧K} (M : Model R K C toK) : AlgebraicGeometry.PrestableFamily M.toBase ↔ M.IsProper ∧ AlgebraicGeometry.PrestableFamily toK ∧ AlgebraicGeometry.PrestableFamily (specialFiber R M.toBase).hom

A model over a discrete valuation ring is a prestable family exactly when it is proper and its generic curve and its special fibre are prestable curves.