The tangent cone of a node #
Let P be a regular local ring whose maximal ideal is generated by two coordinates s₀, s₁ with
dual derivations Dᵢ, for instance the local ring at a closed point of a smooth surface over a
field, and let A = P ⧸ (f) for some f ∈ 𝔪_P² whose Hessian determinant is a unit. By
TauCeti.exists_hypersurface_presentation_of_formallyUnramified, the local ring at a node of a
nodal curve over a field is of this form. This file shows that A satisfies the hypotheses of
the computation in TauCeti.RingTheory.Node.ExceptionalFiber. Hence the fibre of the blowup of
Spec A at its closed point is an étale algebra of dimension two over the residue field κ: the
two tangent directions at the node, the directions of the two branches through it, form a finite
étale κ-scheme of degree two, which need not split over κ.
Write f = a s₀² + b s₀ s₁ + c s₁² (Ideal.mem_span_pair_sq). The Hessian condition says that
b² - 4ac is a unit (TauCeti.isUnit_det_hessian_iff_isUnit_discrim). A change of generators
makes the coefficient of the square of the second generator a unit: swap s₀ and s₁ when a
is a unit, and otherwise replace s₀ by s₀ + s₁, since when a, c ∈ 𝔪 the coefficient b
and hence a - b + c are units. Dividing by that coefficient gives the equation
a x² + b x y + y² with b² - 4a a unit
(TauCeti.exists_eq_unit_mul_quadratic_of_isUnit_discrim). The generator x is then a
nonzerodivisor of A: since P is regular, (x) is prime, and it does not contain
a x² + b x y + y² because it does not contain y
(TauCeti.IsLocalRing.notMem_span_singleton_of_span_pair_eq_maximalIdeal and
TauCeti.IsRegularLocalRing.apply_mem_nonZeroDivisors_of_ker_eq_span). The maximal ideal of A
is not principal, since f ∈ 𝔪_P² imposes no relation on the cotangent plane of P
(TauCeti.IsLocalRing.not_isPrincipal_maximalIdeal_of_ker_le_sq). Finally
y² = -x (a x + b y) lies in x 𝔪_A, so the chart Spec A[𝔪/x] is the whole blowup
(TauCeti.AlgebraicGeometry.isIso_affineBlowupι_of_sq_mem).
At a node, the regularity of P follows from TauCeti.IsRegularLocalRing.of_formallySmooth,
since the local ring P produced by
TauCeti.exists_hypersurface_presentation_of_formallyUnramified is formally smooth and essentially
of finite type over the base field.
Main results #
TauCeti.exists_quadratic_relation_of_isUnit_det_hessian: generatorsx, yof𝔪_Awith a relationa x² + b x y + y² = 0,b² - 4aa unit andxa nonzerodivisor, as required byTauCeti.adjoinRootEquivTensorAffineBlowupandTauCeti.etale_tensor_affineBlowup.TauCeti.exists_etale_tensor_affineBlowup_of_isUnit_det_hessian: the fibre of the blowup ofSpec Aat its closed point is an étaleκ-algebra of dimension two.TauCeti.exists_etale_tensor_affineBlowup_of_formallyUnramifiedandAlgebra.Presentation.exists_etale_tensor_affineBlowup_of_formallyUnramified: the same at a node, for a quotient of a formally smooth local ring and for a presented curve.
References #
- Stacks Project, Section 53.19, Tag 0C46: nodes of curves, defined by a nondegenerate quadratic form in two variables.
- Stacks Project, Section 10.70, Tag 052P: blowup algebras.
A plane curve singularity with nondegenerate quadratic part. Let P be a regular local
ring whose maximal ideal is generated by s₀, s₁, with derivations Dᵢ dual to them, and let
A = P ⧸ (f) for some f ∈ 𝔪_P² whose Hessian determinant is a unit. Then the maximal ideal
of A is generated by two elements x, y with a x² + b x y + y² = 0 for some a, b with
b² - 4a a unit, x is a nonzerodivisor, and 𝔪_A is not principal.
The exceptional fibre of blowing up a plane curve singularity with nondegenerate quadratic
part. Under the hypotheses of TauCeti.exists_quadratic_relation_of_isUnit_det_hessian, the
maximal ideal of A is generated by two elements x, y with y² ∈ x 𝔪_A, so that the chart
Spec A[𝔪/x] is the whole blowup of Spec A at its closed point, and the fibre
κ ⊗[A] A[𝔪/x] of this blowup over the closed point is an étale κ-algebra of dimension two.
The exceptional fibre of blowing up a node. Let A = Q ⧸ (g₁, …, g_c) be as in
TauCeti.exists_hypersurface_presentation_of_formallyUnramified: Q is a local K-algebra,
essentially of finite type and formally smooth over a field K, with Ω[Q⁄K] free of rank
c + 1, and A is local with A ⧸ Fitt₁(Ω[A⁄K]) unramified over K and Fitt₁(Ω[A⁄K]) proper,
as at a node of a nodal curve over K. Then the maximal ideal of A is generated by two elements
x, y with y² ∈ x 𝔪_A, so that the chart Spec A[𝔪/x] is the whole blowup of Spec A at its
closed point, and the fibre κ ⊗[A] A[𝔪/x] of this blowup over the closed point is an étale
κ-algebra of dimension two.
The exceptional fibre of blowing up a node of a presented curve. Let
S = K[x₁, …, x_{c+1}] ⧸ (f₁, …, f_c) be presented over a field K by c + 1 generators and c
relations, with S ⧸ Fitt₁(Ω[S⁄K]) unramified over K, and let q be a prime of S containing
Fitt₁(Ω[S⁄K]). Then the maximal ideal of the local ring S_q is generated by two elements
x, y with y² ∈ x 𝔪, so that the chart Spec S_q[𝔪/x] is the whole blowup of Spec S_q at its
closed point, and the fibre κ ⊗ S_q[𝔪/x] of this blowup over the closed point is an étale
κ-algebra of dimension two.