The local ring at a node is a plane curve singularity #
Let K be a field and let A = Q ⧸ (g₁, …, g_c) be a quotient of a local K-algebra Q,
essentially of finite type and formally smooth over K, whose differentials Ω[Q⁄K] are free of
rank c + 1: for instance the local ring at a point of a curve cut out by c equations in affine
(c + 1)-space. Suppose that the singular locus of A, cut out by the first Fitting ideal of
Ω[A⁄K], is unramified over K and contains the closed point, as at a node of a nodal curve over
K. This file shows that A is then a plane curve singularity whose quadratic part is
nondegenerate: A ≅ P ⧸ (f), where P is a formally smooth local K-algebra, the local ring of
a smooth surface, with coordinates y₁, y₂ (they generate 𝔪_P, and dy₁, dy₂ is a basis of
Ω[P⁄K]), f ∈ 𝔪_P², and the Hessian determinant of f with respect to y₁, y₂ is a
unit. Writing f = a y₁² + b y₁ y₂ + c y₂², the last condition says that b² - 4ac is a unit
(TauCeti.isUnit_det_hessian_iff_isUnit_discrim), so that the quadratic part of f cuts out two
distinct tangent lines, possibly conjugate over a separable quadratic extension of the residue
field. These are the tangent directions of the two branches through the node.
The smoothness of Q is essential: freeness of Ω[Q⁄K] alone does not make Q regular in
positive characteristic. For instance, in characteristic p, the quotient of
K[x, y, z]_(x, y, z) ⧸ (z ^ p) by z - x ^ p and x y is K[x, y]_(x, y) ⧸ (x ^ (p ^ 2), x y).
It satisfies the hypotheses on A without being a node, since the corresponding P is the
nonreduced ring K[x, y]_(x, y) ⧸ (x ^ (p ^ 2)).
The proof assembles three facts: the residue field at a node is separable over K, so Q has
coordinates (TauCeti.exists_coordinates); the cotangent space at a node is a plane
(TauCeti.finrank_cotangentSpace_eq_two), so A is a hypersurface in a local ring P with two
coordinates (TauCeti.exists_hypersurface_presentation), which is formally smooth because the
differentials of the relations cutting it out extend to a basis of Ω[Q⁄K]
(TauCeti.formallySmooth_quotient_of_basis_kaehlerDifferential); and the nodal criterion forces
the Hessian of that hypersurface to be nondegenerate
(TauCeti.mem_maximalIdeal_sq_and_isUnit_det_hessian_of_formallyUnramified).
A presentation S = K[x₁, …, x_{c+1}] ⧸ (f₁, …, f_c), such as a standard syntomic presentation of
relative dimension one, localizes at every prime q of S to such a quotient, with Q the
localization of the polynomial ring, which is formally smooth. So the theorem applies at every
point of the singular locus of S when that singular locus is unramified over K.
Main results #
TauCeti.exists_hypersurface_presentation_of_formallyUnramified: the local ring at a node of a quotient ofQbycrelations is a plane curve singularity with nondegenerate quadratic part.Algebra.Presentation.exists_hypersurface_presentation_of_formallyUnramified: the same for the localization at a singular point of an algebra presented byc + 1generators andcrelations.
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.
The local ring at a node is a plane curve singularity. Let Q be a local K-algebra,
essentially of finite type and formally smooth over a field K, whose differentials Ω[Q⁄K] have
a basis indexed by ι, and let π : Q → A be a surjection whose kernel is generated by |ι| - 1
relations g : σ → Q. Suppose that A ⧸ Fitt₁(Ω[A⁄K]) is unramified over K and that
Fitt₁(Ω[A⁄K]) is a proper ideal. Then all relations but one, g s₀, cut out a formally smooth
local ring P = Q ⧸ (g s : s ≠ s₀) with coordinates y : Fin 2 → Q: the images of the yⱼ
generate 𝔪_P and their differentials form a basis of Ω[P⁄K]. The image f of g s₀ in P
lies in 𝔪_P² and has a unit Hessian determinant, and π induces a K-algebra isomorphism
P ⧸ (f) ≃ A.
The local ring at a node of a presented curve is a plane curve singularity. 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]). Let Q be the localization of the polynomial ring at the inverse image of q,
and g s the image in Q of the relation f_s. Then all relations but one, g s₀, cut out a
formally smooth local ring P = Q ⧸ (g s : s ≠ s₀) with coordinates y : Fin 2 → Q, the image
f of g s₀ in P lies in 𝔪_P² and has a unit Hessian determinant, and P ⧸ (f) is
isomorphic to the localization of S at q, compatibly with the maps from the polynomial
ring.