Nodality of syntomic curves is detected on fibres #
A morphism syntomic of relative dimension one is at worst nodal exactly when each of its
fibres over a residue field Spec κ(y) is at worst nodal. Equivalently, it suffices that every
base change to a field is at worst nodal.
The singular-locus half of the nodal criterion is fibrewise: the formation of the relative
singular subscheme Sing(X/S) commutes with base change
(AlgebraicGeometry.Scheme.isPullback_singularLocus_subschemeι), and a morphism locally of finite
type is formally unramified as soon as its base changes to fields are
(AlgebraicGeometry.Scheme.Hom.formallyUnramified_of_formallyUnramified_baseChange_field). Hence
the singular subscheme Sing(X/S) → S is unramified exactly when every singular subscheme
Sing(X_s/κ(s)) → Spec κ(s) is.
Over a field k, the criterion asks that X → Spec k be syntomic of relative dimension one and
that its singular subscheme, cut out by the first Fitting ideal of Ω_{X/k}, be unramified over
k. Thus the theorem below reduces nodality of a syntomic family to nodality of the curves over
the residue fields of its base.
Syntomicity is itself detected on fibres for flat morphisms locally of finite presentation
(TauCeti.AlgebraicGeometry.syntomicOfRelativeDimension_iff_flat_and_fiberToSpecResidueField).
Combining the two gives the fibrewise criterion: a morphism is at worst nodal of relative dimension
one exactly when it is flat and locally of finite presentation and its fibres over the residue
fields are at worst nodal curves. Such fibres have pure dimension one
(TauCeti.AlgebraicGeometry.PureRelativeDimension.of_syntomicOfRelativeDimension).
Over a local domain of Krull dimension at most one, for instance a discrete valuation ring, every point of the base is the generic or the closed point, so the fibrewise criterion reduces to the generic and the special fibre.
Main results #
AtWorstNodalOfRelativeDimensionOne.of_baseChange_field: a syntomic morphism of relative dimension one is at worst nodal if all its base changes to fields are.atWorstNodalOfRelativeDimensionOne_iff_syntomic_and_fiberToSpecResidueField: a morphism is at worst nodal of relative dimension one exactly when it is syntomic of relative dimension one and all its fibres over residue fields are at worst nodal.atWorstNodalOfRelativeDimensionOne_iff_flat_and_fiberToSpecResidueField: a morphism is at worst nodal of relative dimension one exactly when it is flat and locally of finite presentation and all its fibres over residue fields are at worst nodal.atWorstNodalOfRelativeDimensionOne_iff_genericFiber_and_specialFiber: over a local domain of dimension at most one, a morphism is at worst nodal exactly when it is flat and locally of finite presentation and its generic and special fibres are at worst nodal.
References #
- Stacks Project, Lemma 53.20.1, Tag 0C58: the equivalence of the fibrewise definition of families of nodal curves with the singular-locus criterion.
- The Stacks Project, Tag 02G8 (the fibre criterion for unramified morphisms).
A morphism syntomic of relative dimension one is at worst nodal if its base change along every field-valued point of the target is at worst nodal.
The fibrewise criterion for nodality of syntomic curves. A morphism is at worst nodal of
relative dimension one exactly when it is syntomic of relative dimension one and each of its fibres
f.fiber y ⟶ Spec κ(y) over a residue field is at worst nodal.
The fibrewise criterion for nodal curves. A morphism is at worst nodal of relative
dimension one exactly when it is flat and locally of finite presentation and each of its fibres
f.fiber y ⟶ Spec κ(y) over a residue field is at worst nodal.
The two-fibre criterion for nodal curves. Over a local domain of dimension at most one, for instance a discrete valuation ring, a morphism is at worst nodal of relative dimension one exactly when it is flat and locally of finite presentation and both its generic fibre over the fraction field and its special fibre over the residue field are at worst nodal.