The quadratic part of a hypersurface at a node

Let P be a local K-algebra with coordinates x : ι → P generating its maximal ideal 𝔪, whose differentials dxᵢ form a basis of Ω[P⁄K]: for instance the local ring at a closed point of a smooth variety, with coordinates centred there. Let f ∈ 𝔪, so that A = P ⧸ (f) is the local ring at a point of the hypersurface f = 0. With n = |ι|, the (n - 1)-st Fitting ideal of Ω[A⁄K] is generated by the partial derivatives ∂f/∂xᵢ (TauCeti.fittingIdeal_kaehlerDifferential_quotient_span_singleton). When P is the local ring at a closed point of a smooth n-dimensional variety over a field K and f ≠ 0, so that A is the local ring of a hypersurface of dimension n - 1, this Fitting ideal cuts out the singular locus of A; the results below are stated for the Fitting ideal itself.

The scheme-theoretic criterion for a node asks that this singular locus be unramified over K and pass through the closed point. This file shows that it forces the quadratic part of f to be nondegenerate: f ∈ 𝔪², and the Hessian determinant det (∂²f/∂xᵢ∂xⱼ) is a unit, that is, the polar form of the quadratic part of f is nondegenerate. More precisely, the closed subscheme cut out by the Fitting ideal is the reduced closed point exactly when these two conditions hold. For a plane curve, n = 2, and writing f = a s² + b s t + c t² with a, b, c ∈ P, the second condition says that the discriminant b² - 4ac is a unit (TauCeti.isUnit_det_hessian_iff_isUnit_discrim). Geometrically, the quadratic part of f then cuts out two distinct lines in the tangent plane, possibly conjugate over a separable quadratic extension of the residue field; they are the tangent directions of the two branches through the node, which this file does not construct.

The same holds over an arbitrary base, at a point that need not be closed. Let W be an R-algebra whose differentials have a basis indexed by ι, with dual partial derivatives ∂ᵢ; for instance a smooth relative surface over R on which Ω[W⁄R] is free, containing a relative plane curve f = 0. If the Fitting-ideal locus of W ⧸ (f) is unramified over R, then at every prime 𝔭 of W lying in it, f is critical (∂ᵢ f ∈ 𝔭) and its Hessian determinant is not in 𝔭. When moreover W is smooth over R and the basis is indexed by Fin 2, this is the input that Module.Basis.exists_etale_eq_algebraMap_add_quadratic needs to put f, étale locally around 𝔭, in the form h + a s² + b s t + c t². The local statement above is the special case where W = P is a local K-algebra and 𝔭 = 𝔪.

Main results

• TauCeti.comap_fittingIdeal_kaehlerDifferential_eq_maximalIdeal_iff: the closed subscheme of f = 0 cut out by the (n - 1)-st Fitting ideal of its differentials is the reduced closed point exactly when f ∈ 𝔪² and the Hessian determinant of f is a unit.
• Module.Basis.forall_mem_and_det_hessian_notMem_of_formallyUnramified: over any base R, if the Fitting-ideal locus of W ⧸ (f) is unramified, then at every prime 𝔭 in it the partial derivatives of f lie in 𝔭 and its Hessian determinant does not.
• TauCeti.mem_maximalIdeal_sq_and_isUnit_det_hessian_of_formallyUnramified: if this closed subscheme of P ⧸ (f) is unramified and contains the closed point, then f ∈ 𝔪² and the Hessian determinant of f is a unit.

References

• Stacks Project, Section 53.19, Tag 0C46: nodes of curves, defined by a nondegenerate quadratic form in two variables.
• Stacks Project, Lemma 53.20.1, Tag 0C58: the singular-locus criterion for families of nodal curves.

theorem Module.Basis.forall_mem_and_det_hessian_notMem_of_formallyUnramified {A : Type u_1} {W : Type u_2} [CommRing A] [CommRing W] [Algebra A W] {ι : Type u_3} [Fintype ι] [DecidableEq ι] [Nonempty ι] (b : Basis ι W Ω[W⁄A]) (f : W) [Module.Finite (W ⧸ Ideal.span {f}) Ω[W ⧸ Ideal.span {f}⁄A]] [Algebra.FormallyUnramified A ((W ⧸ Ideal.span {f}) ⧸ TauCeti.fittingIdeal (W ⧸ Ideal.span {f}) Ω[W ⧸ Ideal.span {f}⁄A] (Fintype.card ι - 1))] (P : Ideal W) [P.IsPrime] (hP : Ideal.comap (Ideal.Quotient.mk (Ideal.span {f})) (TauCeti.fittingIdeal (W ⧸ Ideal.span {f}) Ω[W ⧸ Ideal.span {f}⁄A] (Fintype.card ι - 1)) ≤ P) : (∀ (i : ι), ((b.coord i).compDer (KaehlerDifferential.D A W)) f ∈ P) ∧ (TauCeti.hessian (fun (i : ι) => (b.coord i).compDer (KaehlerDifferential.D A W)) f).det ∉ P

A node of a hypersurface over any base has nondegenerate quadratic part. Let b be a basis of Ω[W⁄A] indexed by ι, with dual partial derivatives ∂ᵢ, and let f ∈ W. Suppose that the closed subscheme of W ⧸ (f) cut out by the (|ι| - 1)-st Fitting ideal of its differentials (the singular locus, when W is smooth over A of relative dimension |ι| and W ⧸ (f) is flat over A of relative dimension |ι| - 1) is unramified over A, and let P be a prime of W whose image lies in it. Then f is critical at P, that is ∂ᵢ f ∈ P for all i, and its Hessian determinant det (∂ᵢ ∂ⱼ f) does not lie in P. The base A is arbitrary, and P need not be maximal.

theorem TauCeti.comap_fittingIdeal_kaehlerDifferential_eq_maximalIdeal_iff {K : Type u_1} {P : Type u_2} {ι : Type u_3} [CommRing P] [IsLocalRing P] [Fintype ι] [DecidableEq ι] [Nonempty ι] {x : ι → P} {f : P} [CommRing K] [Algebra K P] [Module.Finite (P ⧸ Ideal.span {f}) Ω[P ⧸ Ideal.span {f}⁄K]] (b : Module.Basis ι P Ω[P⁄K]) (hx : IsLocalRing.maximalIdeal P = Ideal.span (Set.range x)) (hb : ∀ (i : ι), b i = (KaehlerDifferential.D K P) (x i)) : Ideal.comap (Ideal.Quotient.mk (Ideal.span {f})) (fittingIdeal (P ⧸ Ideal.span {f}) Ω[P ⧸ Ideal.span {f}⁄K] (Fintype.card ι - 1)) = IsLocalRing.maximalIdeal P ↔ f ∈ IsLocalRing.maximalIdeal P ^ 2 ∧ IsUnit (hessian (fun (i : ι) => (b.coord i).compDer (KaehlerDifferential.D K P)) f).det

The Fitting ideal of a hypersurface with nondegenerate Hessian. If the differentials dxᵢ of coordinates generating 𝔪 form a basis of Ω[P⁄K], then the closed subscheme of P ⧸ (f) cut out by the (|ι| - 1)-st Fitting ideal of its differentials (the singular locus, in the smooth setting of the module docstring) is the reduced closed point exactly when f ∈ 𝔪² and the Hessian determinant of f is a unit.

theorem TauCeti.mem_maximalIdeal_sq_and_isUnit_det_hessian_of_formallyUnramified {K : Type u_1} {P : Type u_2} {ι : Type u_3} [CommRing P] [IsLocalRing P] [Fintype ι] [DecidableEq ι] [Nonempty ι] {x : ι → P} {f : P} [CommRing K] [Algebra K P] [Module.Finite (P ⧸ Ideal.span {f}) Ω[P ⧸ Ideal.span {f}⁄K]] (b : Module.Basis ι P Ω[P⁄K]) (hx : IsLocalRing.maximalIdeal P = Ideal.span (Set.range x)) (hb : ∀ (i : ι), b i = (KaehlerDifferential.D K P) (x i)) [Algebra.FormallyUnramified K ((P ⧸ Ideal.span {f}) ⧸ fittingIdeal (P ⧸ Ideal.span {f}) Ω[P ⧸ Ideal.span {f}⁄K] (Fintype.card ι - 1))] (hsing : fittingIdeal (P ⧸ Ideal.span {f}) Ω[P ⧸ Ideal.span {f}⁄K] (Fintype.card ι - 1) ≠ ⊤) : f ∈ IsLocalRing.maximalIdeal P ^ 2 ∧ IsUnit (hessian (fun (i : ι) => (b.coord i).compDer (KaehlerDifferential.D K P)) f).det

The quadratic part of f at a node is nondegenerate. Let the differentials dxᵢ of coordinates generating 𝔪 form a basis of Ω[P⁄K]. If the closed subscheme of P ⧸ (f) cut out by the (|ι| - 1)-st Fitting ideal of its differentials (the singular locus, in the smooth setting of the module docstring) is unramified over K and contains the closed point, then f ∈ 𝔪² and the Hessian determinant of f is a unit. This is the local case of Module.Basis.forall_mem_and_det_hessian_notMem_of_formallyUnramified.