# The equivariant Jacobian Conjecture fails for pseudo-planes

Abstract : A smooth complex variety satisfies the Generalized Jacobian Conjecture if all its etale endomorphisms are proper. We study the equivariant version of the conjecture for ℚ-acyclic surfaces of negative Kodaira dimension and infinite algebraic groups. We show that it holds for groups other than ℂ∗, and for ℂ∗ we classify counterexamples relating them to Belyi-Shabat polynomials. Taking universal covers we get rational simply connected ℂ∗-surfaces of negative Kodaira dimension which admit non-proper ℂ∗-equivariant etale endomorphisms. We prove that for every integers r≥1, k≥2 the ℚ-acyclic rational hyperplane u(1+urv)=wk, which has fundamental group ℤ/kℤ and negative Kodaira dimension, admits families of non-proper etale 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 :
Pré-publication, Document de travail
2017

https://hal.archives-ouvertes.fr/hal-01429264
Soumis le : samedi 7 janvier 2017 - 15:15:23
Dernière modification le : vendredi 8 juin 2018 - 14:50:07

### Identifiants

• HAL Id : hal-01429264, version 1
• ARXIV : 1701.01425

### Citation

Adrien Dubouloz, Karol Palka. The equivariant Jacobian Conjecture fails for pseudo-planes. 2017. 〈hal-01429264〉

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