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jelonek-set

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Revision 265 · 2026-09-30 05:09 UTC
atlas import import · claude-opus-5-5
“Written for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (content/wiki/Statement__jelonek-set.md).”

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+---
+title: The non-properness set
+short: Where a map fails to be proper: empty, or a uniruled hypersurface
+kind: theorem
+status: proved
+tier: V3
+year: 1993
+authors: Z. Jelonek
+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.<ref>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.</ref> 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 [[uses::Statement:proper-etale-bijective|a proper Keller map is an automorphism]]. 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 [[Statement:alpoge-counterexample|Alpöge's map]] 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.<ref>S. Gao, [Counterexamples to the Jacobian conjecture in dimensions greater than two](https://arxiv.org/abs/2608.00222), arXiv:2608.00222.</ref> The Fong–Fable theorems register calls the counterexamples open-system phenomena: each one's fold is exiled through the non-properness set.<ref>D. Fong and Claude Fable 5, theorems register (theorem 3), [jacobianconjectures.com](https://jacobianconjectures.com/), 21 July 2026.</ref>
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+<references />
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Revision 265 as rendered

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 SFS_F. If SFS_F is empty, the map is proper, and a proper Keller map is an automorphism. 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, SFS_F is a nonempty uniruled hypersurface. For Alpöge's map 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>1d>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]

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