Configuration Spaces of the Affine Line and their Automorphism Groups

Abstract

The configuration space C n (X) of an algebraic curve X is the algebraic variety consisting of all n-point subsets Q ⊂ X. We describe the automorphisms of C n (C), deduce that the (infinite dimensional) group Aut C n (C) is solvable, and obtain an analog of the Mostow decomposition in this group. The Lie algebra and the Makar-Limanov invariant of C n (C) are also computed. We obtain similar results for the level hypersurfaces of the discriminant, including its singular zero level. This is an extended version of our paper [39]. We strengthened the results concerning the automorphism groups of cylinders over rigid bases, replacing the rigidity assumption by the weaker assumption of tightness. We also added alternative proofs of two auxiliary results cited in [39] and due to Zinde and to the first author. This allowed us to provide the optimal dimension bounds in our theorems.

Automorphism Groups of Configuration Spaces and Discriminant Varieties

“…We obtain similar results for the level hypersurfaces of the discriminant, including its singular zero level. This is an extended version of our paper [39]. We strengthened the results concerning the automorphism groups of cylinders over rigid bases, replacing the rigidity assumption by the weaker assumption of tightness.…”

Infinite transitivity and special automorphisms

“…When this paper was already written, I found the same result in [28, Theorem 1.3], cf. also [32,Theorem 4.10 (a)].…”

Automorphisms in Birational and Affine Geometry

“…For C and P 1 , where the automorphism groups Aff C and PSL(2, C) have dimension 2 and 3, respectively, the problem becomes more difficult. 4 In [39] we used the notation B n for the Artin braid group on n strands. Here we prefer the notation A n−1 that indicates the place of this group among the Artin-Brieskorn groups of series A.…”

“…This preprint is an extended version of our paper [39]. Sections 2-4 of [39] have been modified so that the results stated in [39] under the rigidity assumption are proven now under a weaker assumption of tightness. New Sections 6-9 are devoted to alternative proofs of some of the aforementioned results.…”