# Tangent sweeps

- kind: theorem
- status: proved (evidence tier V4)
- year: 2026
- authors: D. Speyer (the tangent-sweep account, 23 July 2026); S. Gao (arXiv:2608.00222, 31 July 2026), with Claude Fable 5 assisting in the proofs and the write-up
- in one line: Sweeping a curve's tangent lines gives Keller maps; Gao extends this to every n ≥ 3

**Statement.** For every $n\ge3$ there are non-automorphic Keller maps of $\mathbb C^n$ of arbitrarily large geometric degree, built by sweeping tangent directions along hypersurfaces; five explicit examples are given, of $(n,d)=(3,4),(4,5),(4,10),(5,6),(5,12)$.

Speyer found the geometric reason Gallagher's construction works. Take a plane curve $w\mapsto(p(w),q(w))$ and sweep its tangent lines: $(w,\gamma)\mapsto(p+\gamma p',\ q+\gamma q')$. The Jacobian of this sweep is $\gamma(p'q''-p''q')$, which vanishes at the flexes. Reparametrizing so that $q'/p'=w$ gives $(p(w)+\gamma,\ q(w)+\gamma w)$, with Jacobian $\gamma$. A general point lies on as many tangent lines as the class of the curve, the degree of its dual. That number is the number of preimages.[1]

Gao generalized the mechanism from plane curves to direction fields on hypersurfaces. The result is counterexamples in every dimension $n\ge3$, with arbitrarily large geometric degree in each fixed dimension, and five new explicit maps checked in exact rational arithmetic.[2] So [the tangent-sweep theorem refutes JC₃ and every higher dimension, by a mechanism rather than by a single certificate](/wiki/Statement:jc-3.md). It is also [the account of why Alpöge's map works, whose three preimages are the three tangent lines through a point](/wiki/Statement:alpoge-counterexample.md).

In Gao's indexing, a seed of degree $d$ gives geometric degree $d+1$, and $d+1$ is the class of the curve. His four-dimensional maps $F_4$ (degree 5, determinant $-44/9$) and $F_5$ (degree 10, determinant $160/29$) are proved inequivalent to $\Phi\times\mathrm{id}$ only for $\Phi$ of a different geometric degree. Whether $F_4$ is a degree-5 Gallagher map padded with the identity is left open, which bears on the [stabilization question](/wiki/Statement:stabilization-question.md). Gao's Open Problem 4.8 asks for rigidity of the construction's middle branch.

## Relations

- refutes [jc-3](/wiki/Statement:jc-3.md) — The result is counterexamples in every dimension $n\ge3$, with arbitrarily large geometric degree in each fixed dimension, and five new explicit maps checked in exact rational arithmetic. So the tangent-sweep theorem refutes JC₃ and every higher dimension, by a mechanism rather than by a single certificate. [2]
- related to [alpoge-counterexample](/wiki/Statement:alpoge-counterexample.md) — It is also the account of why Alpöge's map works, whose three preimages are the three tangent lines through a point.

## References

1. D. Speyer, comment of 23 July 2026 on [The new counterexample to the Jacobian conjecture](https://sbseminar.wordpress.com/2026/07/20/the-new-counterexample-to-the-jacobian-conjecture/), Secret Blogging Seminar.
2. S. Gao, [Counterexamples to the Jacobian conjecture in dimensions greater than two](https://arxiv.org/abs/2608.00222), arXiv:2608.00222 (31 July 2026).

## Relations to this page

- related to [stabilization-question](/wiki/Statement:stabilization-question.md) — This question is left open by Gao's tangent-sweep paper for its own four-dimensional maps.

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