# The Jacobian Conjecture fails for pseudo-planes

* Auteur correspondant
Abstract : A smooth complex variety satisfies the Generalized Jacobian Conjecture if all its étale endomorphisms are proper. We study the conjecture for $\mathbb{Q}$-acyclic surfaces of negative logarithmic Kodaira dimension. We show that $G$-equivariant counterexamples for infinite group $G$ exist if and only if $G=\mathbb{C}^*$ and we classify them relating them to Belyi–Shabat polynomials. Taking universal covers we get rational simply connected $\mathbb{C}^*$ -surfaces of negative logarithmic Kodaira dimension which admit non-proper $\mathbb{C}^*$-equivariant étale endomorphisms. We prove also that for every integers $r\geq 1, k\geq 2$ the $\mathbb{Q}$-acyclic rational hyperplane $u(1+u^{r}v)=w^k$, which has fundamental group $\mathbb{Z}_k$ and negative logarithmic Kodaira dimension, admits families of non-proper étale endomorphisms of arbitrarily high dimension and degree, whose members remain different after dividing by the action of the automorphism group by left and right composition.
Keywords :
Type de document :
Article dans une revue
Advances in Mathematics, Elsevier, 2018, 339, pp.248-284. 〈10.1016/j.aim.2018.09.020〉

https://hal.archives-ouvertes.fr/hal-01429264
Soumis le : samedi 7 janvier 2017 - 15:15:23
Dernière modification le : jeudi 20 décembre 2018 - 10:05:14

### Citation

Adrien Dubouloz, Karol Palka. The Jacobian Conjecture fails for pseudo-planes. Advances in Mathematics, Elsevier, 2018, 339, pp.248-284. 〈10.1016/j.aim.2018.09.020〉. 〈hal-01429264〉

### Métriques

Consultations de la notice