The geometric origin, unpacked

The geometric recipe behind the Jacobian counterexample

A prompt-driven mathematical investigation by Alexis Gallagher

The compressed construction starts with an unordered triple of points on a projective line, marks one member of the triple, and then forgets the mark. Normally there are three possible marks. The ingenious step is to remove carefully chosen boundary sets so that both the marked-triple space and the unmarked-triple space become ordinary affine three-space, while the forgetful map remains locally invertible but globally three-to-one.

In plain language: the counterexample is the map “forget which root was marked,” written in exceptionally well-chosen affine coordinates.

The explicit polynomial map arose after Akhil Mathew suggested the problem to Levent Alpöge, who announced the resulting counterexample; the announcement credits Fable with producing it. The provenance of the compressed geometric recipe reproduced below is not established here; this page assesses its mathematics, not its authorship.

This page explains every phrase in the geometric sketch and then gives the coordinate calculation hidden inside it. It assumes familiarity with polynomials and derivatives, not with algebraic geometry.

The whole recipe, translated once

The supplied sketch is:

Take π: P¹ × Sym²(P¹) → Sym³(P¹), (p, {q,r}) ↦ {p,q,r}.
R be its ramification divisor;
H ⊂ Sym³(P¹) ≅ P³ be hyperplane tangent but not osculating
  to the small diagonal;
X := (P¹ × Sym²(P¹)) \ (R ∪ π⁻¹(H)) ≅ A³;
Y := Sym³(P¹) \ H ≅ A³.
π|X: X → Y is counterexample.

1. Encode a cubic by its three roots. Repetitions are allowed and order is ignored, so an unordered triple is a point of Sym³(P¹) ≅ P³.
2. Mark one root. A marked cubic is a point (p,{q,r}) ∈ P¹×Sym²(P¹).
3. Forget the mark. The map π sends (p,{q,r}) to {p,q,r}. A cubic with three distinct roots has three preimages, one for each possible marked root.
4. Remove the collision locus. Delete the points where the marked root is also one of the other two roots. That is the ramification divisor R.
5. Choose and remove one special hyperplane. Its contact with the all-roots-equal curve has multiplicities 2+1, not 3.
6. Use the resulting affine coordinates. The target complement is visibly A³. The source complement also turns out to be A³; this is the non-obvious calculation.

Where the compressed sketch hides work. The identifications Sym²(P¹)≅P², Sym³(P¹)≅P³, and P³\H≅A³ are standard. The assertion X≅A³ for this particular H is the pivotal coordinate lemma. It should not be read as automatic.

1. Projective points become polynomial roots

An ordinary cubic in one variable is determined, up to an overall nonzero scale, by its three roots on P¹, including roots at infinity. Equivalently, use a binary cubic

The four coefficients [a:b:c:d], considered up to common scale, form P³. Factoring the binary cubic produces an unordered triple of projective roots. This is the concrete reason

The same argument with binary quadratics gives Sym²(P¹)≅P². Thus the space of “one marked point plus two unmarked points” is P¹×P², a three-dimensional projective variety.

2. The map π only forgets the marked root

This operation is polynomial in the binary-form coefficients: multiply the linear factor for p by the quadratic whose roots are q,r. For three distinct roots, it has exactly three inverse images. Nothing mysterious has happened yet; it is the familiar fact that a cubic has three roots, rephrased geometrically.

But the count changes at collisions. For the multiset {p,p,r}, “mark the first p” and “mark the second p” describe the same point. This is where the map can cease to look locally one-to-one.

3. R removes precisely the bad marked roots

The ramification divisor R consists of marked triples for which the distinguished point repeats:

In elementary polynomial language, if the marked root is the ordinary number t and the cubic is Φ(T), then

Deleting R therefore keeps only simple marked roots. At every remaining point, forgetting the mark is locally invertible. In algebraic-geometric language, the restricted map is étale.

This does not make it globally one-to-one. Over a cubic with three distinct roots there are still three separated choices of marked root. The entire construction exploits that difference between local invertibility and global injectivity.

4. “Tangent but not osculating” in one calculation

The small diagonal is the curve of triples in which all three points coincide:

Under binary-cubic coordinates it is the twisted cubic. For p=[u:v], a cubic with a triple root at p is

A projective coordinate change can move the tangency point to infinity and the remaining intersection to zero. We may therefore choose H to be the hyperplane where the coefficient of U²V is zero. On the small diagonal that coefficient is −3v²u. Its zero pattern is therefore

The double zero at [1:0]=∞ says that H is tangent to the small diagonal there. It is not a triple zero, so the hyperplane is not osculating: it matches the curve to first order, but not to second order. The remaining simple intersection is at zero.

This 2+1 contact is not decorative terminology. It is the special choice that makes the source complement below collapse to affine three-space.

5. The coordinate calculation hidden in X≅A³ and Y≅A³

The target is the easy half

Outside H, the coefficient of U²V is nonzero. Because binary cubics are defined only up to overall scale, normalize that coefficient to −2. Every point of Y=P³\H then has one and only one representative

The three unrestricted coefficients (A,B,C) are affine coordinates. Thus Y≅A³.

The source is the substantive half

Use affine coordinates (x,y,z) and put u=1+xy. Mark the projective root

When x≠0, this is the ordinary root t=u/x=y+1/x. When x=0, it is the point at infinity. Define the unmarked cubic by the following three coefficients:

Direct expansion gives the homogeneous root identity

So xU-uV divides the binary cubic. The quotient is a binary quadratic, whose two roots are the unmarked pair {q,r}. This turns every (x,y,z) into a point of P¹×Sym²(P¹), and forgetting the marked root produces exactly the displayed polynomial map F(x,y,z)=(A,B,C).

Why does this parametrize all of X, with no duplication? Suppose first that the marked root t is finite. Set

Because R was deleted, the marked root is simple, so D≠0. These formulas recover one and only one (x,y,z) wherever the marked root is finite.

To see that the inverse remains regular at infinity, use the reciprocal root coordinate p=[1:s]. The equation Φ(1,s)=0 gives C=2s−Bs²+2As³. Put

Here dΦ(1,s)/ds=−2δ, so deleting R makes δ invertible. At infinity, s=0, these formulas specialize to x=0, y=B, and z=A−4B². The finite-root and reciprocal-root charts cover P¹, and their formulas agree on the overlap. They are the promised regular inverse, proving X≅A³ rather than merely giving a bijection of points.

The role of the two deletions. Deleting π⁻¹(H) makes the normalization to −2U²V legal. Deleting R makes D nonzero, so the inverse coordinate x=2/D is legal. Both are visible in the formulas.

6. Why the restricted map is a counterexample

After identifying X and Y with affine three-space, π|X is the polynomial map F=(A,B,C):ℂ³→ℂ³ above. It has the two properties that the Jacobian conjecture said could not coexist:

1. Everywhere locally invertible. Removing R removes every ramification point. In affine coordinates this says the Jacobian determinant never vanishes. Direct calculation gives the stronger exact identity det JF=−2.
2. Not globally injective. A generic unmarked cubic has three distinct roots, and any one of them can be the marked root. Therefore a generic target has three preimages.

There is no contradiction with the ordinary inverse function theorem: that theorem is local. Nor is there a contradiction with the original projective map being finite. Once R∪π⁻¹(H) is deleted, the restriction is no longer proper. As roots collide, the corresponding affine source point can escape to infinity instead of becoming a finite critical point.

7. The familiar collision is literally “mark each root”

At the target (A,B,C)=(-1/4,0,0), the binary cubic is

Its three projective roots are ∞,-1/2,+1/2. The three advertised inputs are simply the three ways to mark those roots:

The collision is therefore not an accidental identity among three enormous polynomials. It is the defining three-fold ambiguity of forgetting which cubic root was marked.

8. Is this related to the follow-on construction?

Yes. At generic fiber degree three, the geometric marked-root construction and the weighted-lift inverse equation are exactly the same mechanism in different coordinates.

On the dense chart C≠0, for a finite marked root t, introduce the rescaled root and target coordinates

Substituting these into Φ(t)=0 converts the cubic root equation into

This is precisely the weighted-lift inverse equation R(w)=wP−Q for the degree-three seed

The escape mechanism also matches exactly. Put

The derivative of the weighted inverse equation is p(w)−P=−g. Thus a repeated inverse root forces g=0, while reconstruction uses x=C/(2g). For C≠0, the would-be branch point is sent to infinity. This is the affine formula for deleting the ramification divisor.

What the follow-on procedure adds

The original construction is naturally cubic: it forgets one marked member of an unordered triple. The follow-on procedure keeps the same one-variable inverse-root and escape-at-infinity mechanism, but replaces the quadratic seed p(w) by a polynomial of degree n−1. Then

has degree n, producing generic fiber degree n while the map still lives in fixed dimension three. This is why the follow-on family can reach every n≥3.

So the safe conclusion is stronger than “the ideas seem similar,” but narrower than “the original construction already proves the whole family”: the n=3 constructions are explicitly equivalent on the relevant affine chart; the n>3 construction is a new fixed-dimension extension of their common inverse-root mechanism.

The main explainer’s weighted-lift section develops the other side of this correspondence and continues from degree three to the all-n family.

Reproduce the coordinate checks

The repository checker verifies the homogeneous root identity, the derivative and inverse formulas, the 2+1 hyperplane contact, the exact weighted-lift change of variables, and the three advertised marked roots:

Further reading for standard background: the Stacks Project on projective space, ramification divisors, and hyperplanes. A short set of university lecture notes gives the binary-form identification Symⁿ(P¹)≅Pⁿ.