Family 047 — Affine cancellation: when adding one coordinate hides the difference

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Family 047 · Algebraic and complex geometry
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-12

1. The question

If multiplying a geometric space by a line produces ordinary affine space, was the original space already affine space? This is Zariski’s affine-space cancellation problem.

The question resembles cancelling a common factor in arithmetic, but an isomorphism of products can mix the factors. Even if $X\times\mathbb A^1$ is isomorphic to $\mathbb A^5$, the new coordinates need not respect the projection onto the added line. Removing that line from the formula is therefore not a legitimate cancellation argument.

Family 047 contains one manuscript claiming an explicit negative answer over the complex numbers in dimension four. It gives a polynomial equation for a fourfold $X$, asserts that one extra variable turns its coordinate ring into a polynomial ring, and asserts that the original ring is not polynomial.

The same equation supplies two related claimed failures: a polynomial that becomes a coordinate only after stabilization, and smooth families all of whose fibers are affine three-space but which are not locally products in the Zariski topology.

2. Background and essential definitions

Affine space and its coordinate ring

Complex affine $n$-space $\mathbb A^n$ has polynomial coordinate ring $\mathbb C[x_1,\ldots,x_n]$. A closed affine variety defined by equations has the quotient ring obtained by imposing those equations.

Algebraic isomorphism means polynomial maps in both directions, with polynomial inverses. This is stronger than being homeomorphic, diffeomorphic, or biholomorphic. A smooth variety can resemble Euclidean space in several ways without having a polynomial coordinate system.

If $A$ is the coordinate ring of $X$, the cylinder $X\times\mathbb A^1$ has ring $A[w]$, where $w$ is an independent variable. Cancellation asks whether

$$A[w]\simeq\mathbb C[x_1,\ldots,x_{n+1}] \quad\Longrightarrow\quad A\simeq\mathbb C[x_1,\ldots,x_n].$$

All isomorphisms here preserve the complex scalars.

Coordinates versus equations defining a simple fiber

A polynomial $f\in\mathbb C[x_1,\ldots,x_n]$ is a coordinate if it can be completed to a polynomial coordinate system:

$$\mathbb C[x_1,\ldots,x_n]=\mathbb C[f,g_2,\ldots,g_n].$$

For example, $x+y^2$ is a coordinate of $\mathbb C[x,y]$, since $(x,y)\mapsto(x+y^2,y)$ has inverse $(u,v)\mapsto(u-v^2,v)$.

A coordinate has affine-space fibers. The reverse implication is a separate problem: knowing the quotient by $(f)$ is polynomial does not, on its own, provide an ambient polynomial automorphism containing $f$.

A polynomial is one-stably a coordinate if it becomes a coordinate after adjoining one independent variable. A residual coordinate over a parameter ring means it is a coordinate after passing to each residue field of that parameter ring. These properties concern stabilization and specialization respectively, and neither should be confused with already having global coordinates.

Families with affine-space fibers

An $\mathbb A^3$-fibration is a morphism whose fibers, including geometric fibers, are affine three-space. Being Zariski-locally trivial is stronger: over each member of an algebraic open cover of the base, the entire family must be isomorphic to the product of that open set with $\mathbb A^3$, compatibly with projection.

The Dolgachev–Weisfeiler question asks when affine-space fibers force this local product structure. It concerns polynomial families, not just whether nearby fibers look alike as differentiable manifolds.

3. Development history and previous partial results

Cancellation is known for the affine line through work of Abhyankar, Heinzer, and Eakin. The complex affine-plane case was established through work of Fujita, Miyanishi, and Sugie, using the geometry of affine surfaces and cylinder-like open subsets.

Positive characteristic has a different history. Asanuma constructed polynomial-fiber rings with cylinders that are polynomial. Gupta showed that suitable examples are not affine three-space, giving counterexamples in dimension three. Her subsequent work established failure in every dimension at least three over fields of positive characteristic.

The higher-dimensional positive-characteristic result needs its own family of examples. Simply adding variables to a threefold whose first cylinder is already affine space would make the original obstruction disappear. This is also a useful warning when interpreting the dimension-four claim here.

Over characteristic zero, additive group actions have been central to recognizing spaces that are not affine space. Makar-Limanov’s work on the Russell cubic supplied a famous example distinguished through its locally nilpotent derivations. That result did not by itself provide the cylinder isomorphism required for affine-space cancellation.

Residual-coordinate methods address the other side of the problem: how adjoining a variable can simplify a polynomial. El Kahoui–Ouali, followed by Dutta–Lahiri, proved one-stability theorems under the relevant one-dimensional parameter hypotheses. M1 uses this established framework and gives explicit coordinate calculations for its example.

Thus two distinct tasks remain in a cancellation counterexample: construct the polynomial cylinder and show the original variety is not polynomial. The manuscript’s main new obstruction is directed at the second task. Gupta’s survey gives the broader history and links to the related coordinate and fibration problems.

4. What the new manuscripts claim

The explicit fourfold

Take five independent variables $p,s,u,F,J$, and abbreviate

$$x=s^2+u^3+p^2F.$$

Define the polynomial

$$H=x^2F-(1+2sx)J-p^2J^2-pu$$

in those five variables, and let

$$A=\mathbb C[p,s,u,F,J]/(H), \qquad X=\operatorname{Spec}A.$$

The symbol $x$ is an abbreviation, not a sixth independent variable. Theorem 1.1 of M1 claims that $A$ is an integral four-dimensional algebra and

$$A[w]\simeq\mathbb C[t_1,t_2,t_3,t_4,t_5], \qquad A\not\simeq\mathbb C[t_1,t_2,t_3,t_4].$$

The subsequent fibration statement makes $X$ smooth. Thus the claimed failure is not produced by a singularity that vanishes under an informal change of variables.

Stable coordinates

Corollary 1.2 claims that $H$ is not a coordinate in the original five-variable polynomial ring, but becomes one in that ring with $w$ adjoined. This is failure of the Stable Coordinate Conjecture in ambient dimension five after one stabilization.

The direction from the fourfold claim is straightforward to interpret: if $H$ were already a coordinate, quotienting by it would leave a polynomial ring in four variables. Its asserted nonpolynomial quotient rules that out.

Two nontrivial affine-space families

M1 also claims that the maps

$$p:X\longrightarrow\mathbb A^1, \qquad (p,H):\mathbb A^5\longrightarrow\mathbb A^2$$

are smooth $\mathbb A^3$-fibrations and neither is Zariski-locally trivial. The source of the second map is ordinary affine space, making the distinction between simple total space and simple family structure particularly clear.

The manuscript states the affine-space condition over residue fields, not only after algebraic closure. These are claimed counterexamples in relative dimension three over the indicated smooth normal bases.

5. Why the advance matters

Importance: 5/5. If correct, the example would give a negative answer to a central characteristic-zero recognition problem for affine space, with consequences for two closely connected conjectures.

Cancellation asks whether a polynomial coordinate system can hide essential geometry when one extra direction is available. An explicit counterexample would demonstrate that it can. It would also show that smoothness, affine-space fibers, and a polynomial cylinder do not together recover the missing global polynomial structure.

The equation makes the claim concrete, but its significance is structural rather than computational. The essential issue is whether every possible coordinate system or additive action can be ruled out on the fourfold, while still constructing one after stabilization.

6. The main idea

Make the cylinder polynomial

Away from $p=0$, the manuscript changes variables so that the equation is linear in one of them. At $p=0$, a separate linear change in $F,J$ makes the reduced polynomial a coordinate. This establishes the claimed residual-coordinate behavior.

The explicit stabilization uses a locally nilpotent derivation $D$. This is a differentiation-like operator such that, for each polynomial $a$, some power $D^k(a)$ vanishes. The exponential series therefore terminates on each element and defines an algebraic action of the additive group.

M1 constructs such an operator with $D(H)=p^3$. Exponentiating with the extra variable shifts the equation to $H+p^3w$. A further polynomial coordinate change, with its inverse supplied in the manuscript, produces the polynomial quotient. Divisibility conditions are retained at $p=0$; an isomorphism only after inverting $p$ would not establish the cylinder theorem.

Assume coordinates exist and extract an impossible action

To distinguish $A$ from a polynomial ring, M1 studies orders of vanishing along $p=0$ in an auxiliary algebra. These define a filtration and an associated graded ring, which keeps the leading terms for that filtration.

If $A$ were polynomial, differentiating in a suitable polynomial coordinate would induce a nonzero homogeneous locally nilpotent derivation on this graded ring, with an invariant of positive degree. The manuscript then seeks to rule out precisely such an action.

It lifts the action to the complement of the zero section in a line bundle, where a relevant ideal becomes principal. There are two distinct measurements: the valuation degree coming from $p$, and the weight from scaling the line-bundle fibers. Keeping them separate converts the hypothetical action into a constrained derivation on an explicit determinant ring.

The final obstruction uses the polynomial Mason–Stothers $abc$ inequality, which bounds degrees in polynomial identities, together with a nonsquare condition in the original fraction field. According to M1, these force the derivation to fix certain generators, while the determinant relation then forces a weight incompatible with the earlier constraint.

Why the fibrations cannot be locally products

If either claimed fibration were Zariski-locally a polynomial family, a theorem of Bass–Connell–Wright would identify its algebra with the symmetric algebra of a projective module over the base. Quillen–Suslin then makes that module free over these polynomial bases. The resulting global coordinates would contradict the nonpolynomiality or noncoordinate conclusion already obtained.

This consequence permits arbitrary polynomial transition maps in the proposed local trivializations; it is not merely a failure to be a vector bundle under preassigned linear transitions.

7. Scope, limitations, and open questions

The cancellation example is a complex fourfold. It does not settle characteristic-zero cancellation for affine three-space. Nor does it automatically yield counterexamples in every larger dimension by adjoining variables: the very first added variable is asserted to make this example polynomial.

The stable-coordinate consequence concerns ambient dimension five. The affine-fibration consequences have relative dimension three and the specific bases $\mathbb A^1$ and $\mathbb A^2$. They do not answer the corresponding relative-dimension-one or relative-dimension-two questions.

All distinctions are algebraic. The manuscript’s assertion that $X$ is not affine four-space means no polynomial isomorphism exists; no broader conclusion about every analytic or topological comparison is needed for its theorem.

These limits leave open recognition questions in smaller dimensions and the search for additional hypotheses that force an affine-space fibration to become a locally trivial bundle.

8. Sources and further reading

M1’s Theorem 1.1 gives the cancellation statement, Corollary 1.2 gives stable-coordinate failure, and Section 7 states the fibration consequences. Its introductory ring table separates the original algebra, its degeneration, and the line-bundle construction used in the obstruction.

The primary works linked in Section 3 explain the positive-characteristic precedent and the residual-coordinate machinery. For orientation, the Zariski cancellation problem provides useful terminology. The key logical distinction is between having affine-space fibers, being a polynomial family, and becoming polynomial after an extra variable is added.

Source manuscripts

• M1. OpenAI, An explicit failure of complex affine-space cancellation (September 23, 2026).

Sources consulted on 12 October 2026 at repository revision fd4aeeb. Download the manuscript citations.