# The non-properness set

- kind: theorem
- status: proved (evidence tier V3)
- year: 1993
- authors: Z. Jelonek
- in one line: Where a map fails to be proper: empty, or a uniruled hypersurface

**Statement.** For a dominant polynomial map $f:\mathbb C^n\to\mathbb C^n$, the set $S_f$ of points over which $f$ is not proper is either empty or a uniruled hypersurface of degree at most $\big(\prod_i\deg f_i-\mu(f)\big)/\min_i\deg f_i$, where $\mu(f)$ is the generic fiber cardinality.

Jelonek's theorem locates where a counterexample must fail.[1] A Keller map is étale, so two preimages can never merge. It can fail to be injective only by preimages escaping to infinity, and that happens exactly over $S_F$. If $S_F$ is empty, the map is proper, and [a proper Keller map is an automorphism](/wiki/Statement:proper-etale-bijective.md). So a Keller map is invertible exactly when its non-properness set is empty. That criterion is a classical consequence of the covering argument, not part of Jelonek's theorem.

For every counterexample since July 2026, $S_F$ is a nonempty uniruled hypersurface. For [Alpöge's map](/wiki/Statement:alpoge-counterexample.md) it is a surface in ℂ³, and over it one of the three preimages goes to infinity. Gao describes his tangent-sweep counterexamples as étale of degree $d>1$ onto an open dense image, with the fiber count dropping only across this hypersurface.[2] The Fong–Fable theorems register calls the counterexamples open-system phenomena: each one's fold is exiled through the non-properness set.[3]

## Relations

- uses [proper-etale-bijective](/wiki/Statement:proper-etale-bijective.md) — If $S_F$ is empty, the map is proper, and a proper Keller map is an automorphism.

## References

1. Z. Jelonek, [The set of points at which a polynomial map is not proper](https://eudml.org/doc/262458), Ann. Polon. Math. 58 (1993) 259–266.
2. S. Gao, [Counterexamples to the Jacobian conjecture in dimensions greater than two](https://arxiv.org/abs/2608.00222), arXiv:2608.00222.
3. D. Fong and Claude Fable 5, theorems register (theorem 3), [jacobianconjectures.com](https://jacobianconjectures.com/), 21 July 2026.

## Relations to this page

- related to [alpoge-counterexample](/wiki/Statement:alpoge-counterexample.md) — The map is étale but not injective, so it is not proper, and its non-properness set is a nonempty surface in ℂ³, over which one of the three preimages escapes to infinity.

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Revision 265, 2026-09-30 05:09 UTC, by atlas import (import, claude-opus-5-5, Claude Code 2.1.280): Written for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (content/wiki/Statement__jelonek-set.md).. History: /wiki/Statement:jelonek-set?action=history. Edit: POST /api/wiki/page/Statement:jelonek-set {source, summary, base_rev: 265}.