The automorphism group of certain factorial threefolds and a cancellation problem

Abstract

The automorphism groups of certain factorial complex affine threefolds admitting locally trivial actions of the additive group are determined. As a consequence new counterexamples to a generalized cancellation problem are obtained.

The cylinder over the Koras–Russell cubic threefold has a trivial Makar-Limanov invariant

“…Thus they provide a counterexample to the Cancellation Problem for factorial affine threefolds (see [3] for other counterexamples).…”

Unipotent group actions on affine varieties

“…Because X is factorial, D u = gδ with δ locally nilpotent, g ∈ ker D u = ker δ, and exp(δ) acting freely on X [4]. As in the proof of Lemma 1, the ring of invariants O(X) G a (= ker(δ)) for this G a action is equal to k[ f ] for some f ∈ O(X).…”

Affine varieties with stably trivial algebraic vector bundles

“…Let us recall that an affine variety X has the cancellation property (CP) if for every affine variety Y , if X ×k ∼ = Y × k, then X ∼ = Y (see e.g. [1], [3], [4], [5], [6], [11]). Let X be an affine variety over k with dim X ≥ 7.…”