On endomorphisms of affine spaces and the Jacobian problem

@misc{borisov2026endomorphisms,
  title = {{On endomorphisms of affine spaces and the Jacobian problem}},
  author = {Alexander Borisov and Ofer Gabber and Adrian Vasiu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05746},
  eprint = {2609.05746},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05746},
  note = {CC BY-NC-ND 4.0},
}

TY  - UNPB
TI  - On endomorphisms of affine spaces and the Jacobian problem
AU  - Alexander Borisov
AU  - Ofer Gabber
AU  - Adrian Vasiu
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.05746
UR  - https://arxiv.org/abs/2609.05746
AB  - Let p be a prime. We provide examples which show that étale endomorphisms of affine planes over an algebraically closed field k of characteristic p can have fibers of arbitrary finite cardinal. Let (l,m)∈ℕ×ℕ^∗. We provide examples of such étale endomorphisms whose images have complements of cardinality l and whose geometric degrees are pm. Several conjectures are disproved, and in particular we provide an analog over k of Kulikov's counterexample to the `Generalized Jacobian Conjecture' of Bass for surfaces. Surjective (resp.\ surjective and non-surjective) counterexamples to Adjamagbo's Jacobian Conjecture over each k are included in dimension $2$ (resp.\ in all dimensions at least $3$); for instance, if p=2, then we show for each m there exist surjective étale endomorphisms of the affine spaces over k of dimension at least $3$ of geometric degree m. If e:X→ X is an endomorphism of a variety over an algebraically closed field K, then we show that there exists n∈ℕ such that Imm(e^n)=Imm(e^n+1) provided either (i) e is quasi-finite or (ii) dim(X)≤ 2 and X∖Imm(e) is finite. We provide examples of endomorphisms of affine planes of dimensions at least $3$ over K whose images have complements of cardinality l and for all n∈ℕ we have Imm(e^n)≠Imm(e^n+1). We prove that all affine moduli schemes of étale endomorphisms of affine spaces with Jacobian matrices of determinant 1 over K are connected and classify all those that are smooth. We prove that the Jacobian Conjecture over ℂ and Adjamagbo's analog of it over k hold for étale endomorphisms of affine spaces that are composites g∘ f, where f is quasi-finite and a locally closed embedding in codimension $1$ outside a specific finite subset and g is a projection that omits one coordinate.
ER  -