Seminar Algebra and Geometry: Adrien Dubouloz (Université de Bourgogne)

Examples of rigid surfaces with infinitely transitive cylinders

A result of Makar-Limanov asserts that if X an affine variety without nontrivial action of the additive group, then there are no additive group actions on the cylinder X x A¹ besides the obvious ones by generic translations on the second factor. In this talk, I will give examples which illustrate how badly this property can fail for higher dimensional cylinders. The recipe will involve, among other ingredients, rigid affine cubic surfaces, some G_a-linearized line bundles and a structure theorem for vector bundles over rational affine surfaces, all mixed together through a variant of the overused Danielewski fiber product trick.

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