Affine modifications and affine hypersurfaces with a very transitive automorphism group

Abstract

International audienc

Simple birational extensions of the polynomial algebra ℂ^{[3]}

“…A complete analog of Theorem 7.2 in [19] cited at the beginning holds if the polynomial p ∈ C [4] is linear with respect to two (and not just one) variables. Indeed (3.24) if the 3-fold X, p = a(x, y)u + b(x, y)v + c(x, y) = 0, in C 4 is smooth and acyclic, then p ∈ C [4] is a variable.…”

“…Generalizing a theorem of A. Sathaye [34] it is proven in [19] that if a surface X = p −1 (0) ⊆ C 3 with p = f u + g ∈ C [3] and f, g ∈ C[x, y] is acyclic (that is, H * (X; Z) = 0), then p is a variable of the polynomial ring C [3] , i.e., a coordinate of an automorphism α ∈ Aut C 3 . Thus X can be rectified, and so is isomorphic to C 2 .…”

“…We start by recalling the notion of affine modification [19]; at the same time, we introduce the notation that will be used throughout the paper.…”

“…We would like to use this opportunity to indicate a flaw in the proof of Theorem 7.2 in [19] (which generalizes the Sathaye theorem mentioned in the introduction). Namely, proving Proposition 7.1 in [19] and carrying induction by the multiplicity of a root x = 0 of the polynomial p, it was forgotten to extend it over all the roots. Instead, one can apply 1.29(a) (cf.…”

“…Our aim in this section is to give a criterion for as when the acyclicity is preserved under such a modification (cf. [19,Thm. 3.1]).…”