Nodal reduction of curves of genus zero

Let R be a discrete valuation ring with fraction field K. The projective line ℙ¹_R is a smooth proper model over R of the projective line ℙ¹_K (TauCeti.Model.projectiveLine). Consequently ℙ¹_K has nodal reduction, indeed good reduction, over R.

A regular proper curve of genus zero over K with a divisor of degree one, for instance with a K-rational point, and with K integrally closed in its function field, is isomorphic to ℙ¹_K (AlgebraicGeometry.Scheme.exists_iso_projectiveLine_of_genus_eq_zero_of_divisor_degree_eq_one). Since nodal reduction only depends on the curve up to isomorphism over K, every such curve has nodal reduction over R, without any extension of R. This is the case of a curve of genus zero whose generic fibre already has a divisor of degree one.

In general a smooth proper geometrically integral curve C of genus zero over K acquires a rational point only after a finite separable extension of K, and one of degree at most two suffices: over it, C becomes ℙ¹ (AlgebraicGeometry.Scheme.exists_finrank_le_two_and_iso_projectiveLine_of_genus_eq_zero). Choosing a place of that extension above the closed point of R, with local ring R', the base change of C has the smooth proper model ℙ¹_{R'}.

Main results

• TauCeti.HasNodalReduction.projectiveLine: ℙ¹_K has nodal reduction over R.
• TauCeti.HasNodalReduction.of_genus_eq_zero_of_divisor_degree_eq_one: a regular proper curve of genus zero over K with a divisor of degree one has nodal reduction over R.
• TauCeti.HasNodalReduction.exists_finrank_le_two_of_genus_eq_zero: a smooth proper geometrically integral curve of genus zero over K has nodal reduction after a finite separable extension of K of degree at most two, with a chosen place above the closed point of R.

References

• The Stacks Project, Chapter 55, Semistable Reduction
• Q. Liu, Algebraic Geometry and Arithmetic Curves, Oxford University Press, 2002, Section 10.1: models of curves.

theorem TauCeti.HasNodalReduction.projectiveLine (R K : Type u) [CommRing R] [IsDomain R] [IsDiscreteValuationRing R] [Field K] [Algebra R K] [IsFractionRing R K] : HasNodalReduction R K (AlgebraicGeometry.projectiveLine K) (AlgebraicGeometry.projectiveLine K ↘ AlgebraicGeometry.Spec ↧K)

The projective line has nodal reduction: ℙ¹_R is a smooth proper model of ℙ¹_K.

theorem TauCeti.HasNodalReduction.of_genus_eq_zero_of_divisor_degree_eq_one {R K : Type u} [CommRing R] [IsDomain R] [IsDiscreteValuationRing R] [Field K] [Algebra R K] [IsFractionRing R K] (C : AlgebraicGeometry.Scheme) [C.Over (AlgebraicGeometry.Spec ↧K)] [AlgebraicGeometry.Scheme.IsCurveOver K] [∀ (y : AlgebraicGeometry.CodimensionOnePoint C), IsDiscreteValuationRing ↑(C.presheaf.stalk ↑y)] (hk : IsIntegrallyClosedIn K ↑C.functionField) (hg : Module.finrank K (AlgebraicGeometry.Scheme.Modules.Cohomology (AlgebraicGeometry.InvertibleSheaf.trivial C).obj 1) = 0) {D : AlgebraicGeometry.SchemeWeilDivisor C} (hD : (AlgebraicGeometry.SchemeWeilDivisor.relativeDegree (C ↘ AlgebraicGeometry.Spec ↧K)) D = 1) : HasNodalReduction R K C (C ↘ AlgebraicGeometry.Spec ↧K)

A curve of genus zero with a divisor of degree one has nodal reduction. Let C be a regular proper curve over the fraction field K of a discrete valuation ring R, with K integrally closed in K(C). If dim_K H¹(C, 𝒪_C) = 0 and some divisor on C has degree one, then C has a proper at-worst-nodal model over R; indeed C ≅ ℙ¹_K, which has the smooth proper model ℙ¹_R.

theorem TauCeti.HasNodalReduction.exists_finrank_le_two_of_genus_eq_zero {R K : Type u} [CommRing R] [IsDomain R] [IsDiscreteValuationRing R] [Field K] [Algebra R K] [IsFractionRing R K] (C : AlgebraicGeometry.Scheme) [C.Over (AlgebraicGeometry.Spec ↧K)] [AlgebraicGeometry.IsProper (C ↘ AlgebraicGeometry.Spec ↧K)] [AlgebraicGeometry.SmoothOfRelativeDimension 1 (C ↘ AlgebraicGeometry.Spec ↧K)] [AlgebraicGeometry.GeometricallyIntegral (C ↘ AlgebraicGeometry.Spec ↧K)] (hg : Module.finrank K (AlgebraicGeometry.Scheme.Modules.Cohomology (AlgebraicGeometry.InvertibleSheaf.trivial C).obj 1) = 0) : ∃ (E : FiniteDVRExtension R K), Module.finrank K E.extensionField ≤ 2 ∧ HasNodalReduction E.localRing E.extensionField (genericFiber K E.extensionField (C ↘ AlgebraicGeometry.Spec ↧K)).left (genericFiber K E.extensionField (C ↘ AlgebraicGeometry.Spec ↧K)).hom

A smooth curve of genus zero has nodal reduction after a separable extension of degree at most two. Let C be a smooth proper geometrically integral curve over the fraction field K of a discrete valuation ring R, with dim_K H¹(C, 𝒪_C) = 0. Then there is a finite separable extension K' of K of degree at most two, with a chosen place above the closed point of R and local ring R', such that C_{K'} has a proper at-worst-nodal model over R'; indeed C_{K'} ≅ ℙ¹_{K'}, which has the smooth proper model ℙ¹_{R'}.