Sections of smooth relative curves are effective Cartier divisors

Let f : X ⟶ S be a morphism of schemes and s : S ⟶ X a section of f, so that s ≫ f = 𝟙 S, which is a closed immersion, as it is whenever f is separated. Its ideal sheaf is s.ker. If s lands in an open subscheme U ⊆ X on which f is smooth of relative dimension one, for instance the smooth locus of a family of nodal curves, then the image of s is an effective Cartier divisor on X, and indeed a relative effective Cartier divisor over S: its closed subscheme is isomorphic to S. Sections through the smooth locus are the markings of pointed curves, and these divisors are the input for the log-canonical sheaf ω_{X/S}(s₁ + ⋯ + sₙ) of a pointed family.

Away from the image of s, the ideal sheaf is the unit ideal. Near a point s y, choose affine opens W ∋ y of S and V ∋ s y of X with s(W) ⊆ V ⊆ f⁻¹(W) on which Γ(S, W) → Γ(X, V) is standard smooth of relative dimension one. The restriction of s is then an algebra retraction Γ(X, V) → Γ(S, W), and Algebra.IsStandardSmoothOfRelativeDimension.exists_isSMulRegular_smul_ker_le_span_singleton gives a nonzerodivisor generating its kernel after inverting a function r that is a unit along the section.

Main results

• AlgebraicGeometry.Scheme.Hom.isEffectiveCartier_ker_of_comp_eq_id: a section which is a closed immersion, through an open subscheme that is smooth of relative dimension one over the base, is an effective Cartier divisor.
• AlgebraicGeometry.Scheme.Hom.isRelativeEffectiveCartier_ker_of_comp_eq_id: it is a relative effective Cartier divisor over the base.
• AlgebraicGeometry.Scheme.Hom.isRelativeEffectiveCartier_ker_of_smoothOfRelativeDimension: the case of a section of a separated smooth relative curve.

References

• The Stacks Project, Divisors, sections Regular immersions and Relative effective Cartier divisors: a section of a smooth morphism of relative dimension one is a relative effective Cartier divisor.

theorem AlgebraicGeometry.Scheme.Hom.isEffectiveCartier_ker_of_comp_eq_id {X S : Scheme} {f : X ⟶ S} (s : S ⟶ X) (hs : CategoryTheory.CategoryStruct.comp s f = CategoryTheory.CategoryStruct.id S) [IsClosedImmersion s] (U : X.Opens) (hU : Set.range ⇑s ⊆ ↑U) [SmoothOfRelativeDimension 1 (CategoryTheory.CategoryStruct.comp U.ι f)] : (ker s).IsEffectiveCartier

A section of a smooth relative curve is an effective Cartier divisor. Let s be a section of f : X ⟶ S which is a closed immersion (as it is when f is separated), and whose image lies in an open subscheme U ⊆ X on which f is smooth of relative dimension one. Then the ideal sheaf of s is locally generated by a nonzerodivisor.

theorem AlgebraicGeometry.Scheme.Hom.isRelativeEffectiveCartier_ker_of_comp_eq_id {X S : Scheme} {f : X ⟶ S} (s : S ⟶ X) (hs : CategoryTheory.CategoryStruct.comp s f = CategoryTheory.CategoryStruct.id S) [IsClosedImmersion s] (U : X.Opens) (hU : Set.range ⇑s ⊆ ↑U) [SmoothOfRelativeDimension 1 (CategoryTheory.CategoryStruct.comp U.ι f)] : (ker s).IsRelativeEffectiveCartier f

A section of a smooth relative curve is a relative effective Cartier divisor. Let s be a section of f : X ⟶ S which is a closed immersion, and whose image lies in an open subscheme U ⊆ X on which f is smooth of relative dimension one. Then the ideal sheaf of s is an effective Cartier divisor whose closed subscheme, isomorphic to S, is flat over S.

theorem AlgebraicGeometry.Scheme.Hom.isRelativeEffectiveCartier_ker_of_smoothOfRelativeDimension {X S : Scheme} {f : X ⟶ S} (s : S ⟶ X) (hs : CategoryTheory.CategoryStruct.comp s f = CategoryTheory.CategoryStruct.id S) [IsSeparated f] [SmoothOfRelativeDimension 1 f] : (ker s).IsRelativeEffectiveCartier f

A section of a separated morphism f : X ⟶ S that is smooth of relative dimension one is a relative effective Cartier divisor.