Étale charts of a node from a split quadratic equation #
Let W be a smooth A-algebra with étale coordinates p, q, that is, elements whose
differentials dp, dq form a basis of Ω[W⁄A], and let P be a prime of W on the section
p = q = 0. Consider the zero locus in Spec W of
f = a p² + b p q + c q² - h, with a, b, c ∈ W and h ∈ A,
where the quadratic form a p² + b p q + c q² is nondegenerate at P (its discriminant
b² - 4ac does not vanish in κ(P)) and split at P (it is a product of two linear forms over
κ(P)). Near a split node, the equation of a nodal curve inside a smooth relative surface takes
this shape once it is normalized to second order; producing that normalization is not done here.
This file shows that, étale locally around P and without changing the residue field, the
quadratic form becomes s t in new étale coordinates s, t vanishing at P. Consequently the
zero locus of f is étale over the node chart A[u, v] ⧸ (uv - h). No hypothesis relates h
to P, so P need not lie on this zero locus; when algebraMap A W h ∈ P it does, and then the
point P' above P maps into the section u = v = 0 of the node chart.
The proof is algebraic. Let ρ be a root of X² + bX + ac with 2ρ + b and ρ + b invertible;
adjoining it to W is a standard étale extension W₁, and a residue root of X² + bX + ac at
P lifts it to a point of W₁ above P with the same residue field. Writing u = ρ + b, one
has u² + ac = bu, hence
(u p + c q) (a p + u q) = u (a p² + b p q + c q²),
so s = u p + c q and t = u⁻¹ (a p + u q) satisfy s t = a p² + b p q + c q². Their
differentials are ds = u dp + c dq and dt = u⁻¹ a dp + dq modulo p and q, with
determinant (u² - ac) / u = 2ρ + b, a unit. Inverting the determinant of ds, dt in the basis
dp, dq makes ds, dt a basis, so A[X, Y] → W' given by X ↦ s and Y ↦ t is étale by the
Jacobian criterion TauCeti.etale_aeval_of_basis. The node chart is its base change along
A[X, Y] → A[X, Y] ⧸ (XY - h) (TauCeti.NodeAlgebra.etale_lift_of_etale_aeval).
Main results #
Module.Basis.exists_etale_mul_eq_quadratic_of_point: the factorization at a field-valued point, allowing the value field to extend the residue field.Module.Basis.exists_etale_mul_eq_quadratic: after an étale extension preserving the residue field atP, the quadratic form is a products tof étale coordinates vanishing atP.TauCeti.NodeAlgebra.etale_lift_of_etale_aeval: ifA[X, Y] → Tgiven byX ↦ sandY ↦ tis étale, then so is the mapA[u, v] ⧸ (uv - h) → T ⧸ (st - h)given byu ↦ sandv ↦ t.Module.Basis.exists_etale_nodeAlgebra_hom_of_point: the node chart at a field-valued point, retaining the equation of the original hypersurface.Module.Basis.exists_etale_nodeAlgebra_hom: étale locally aroundP, the zero locus ofa p² + b p q + c q² - his étale over the node chartA[u, v] ⧸ (uv - h), by a map sendingu, vto elementss, tof the primeP'aboveP.
References #
- Stacks Project, Section 53.20, Tag 0C58: families of nodal curves and their étale local structure.
- Stacks Project, Example 55.14.1, Tag 0CDC: the
local model
uv = πⁿof a node over a discrete valuation ring.
In T ⧸ (xy - a), the images of x and y satisfy the nodal equation.
If R[X, Y] → T given by X ↦ x and Y ↦ y is étale, then so is the map
R[u, v] ⧸ (uv - a) → T ⧸ (xy - a) given by u ↦ x and v ↦ y: it is the base change of the
former along R[X, Y] → R[X, Y] ⧸ (XY - a).
A split nondegenerate quadratic form at a field-valued point is étale locally a product of étale coordinates vanishing at a lift of that point. The value field may extend the residue field, so this also applies to tangent directions which are not rational over the residue field.
A split nondegenerate binary quadratic form in étale coordinates is étale locally a product
of étale coordinates. Let W be a smooth A-algebra and p, q ∈ W elements whose
differentials form a basis of Ω[W⁄A], both lying in a prime P. Let a, b, c ∈ W be such that
the discriminant discrim a b c = b² - 4ac does not vanish at P and X² + bX + ac has a root
in κ(P); that is, the quadratic form a X² + b XY + c Y² is nondegenerate and splits into
linear factors over κ(P). Then there are an étale W-algebra W', a prime P' of W' above
P with the same residue field, and s, t ∈ P' with s t = a p² + b p q + c q², such that
A[X, Y] → W' given by X ↦ s and Y ↦ t is étale.
At a field-valued point where a nondegenerate binary quadratic form splits, its zero locus minus a base parameter is étale locally a node. Both node coordinates vanish at a lift of the point, and their product minus the parameter equals the original equation.
Étale charts of a node from a split quadratic equation. Let W be a smooth A-algebra
and p, q ∈ W elements whose differentials form a basis of Ω[W⁄A], both lying in a prime P.
Let a, b, c ∈ W be such that the quadratic form a X² + b XY + c Y² is nondegenerate and split
at P, as in Module.Basis.exists_etale_mul_eq_quadratic, and let h ∈ A. Then, after an étale
extension W → W' and at a prime P' above P with the same residue field, the zero locus of
f = a p² + b p q + c q² - h admits an étale map to the node chart A[u, v] ⧸ (uv - h): there
are s, t ∈ P' and an étale A-algebra map A[u, v] ⧸ (uv - h) → W' ⧸ (f) sending u ↦ s and
v ↦ t.