Automorphism Groups of Configuration Spaces and Discriminant Varieties

TL;DR

This paper characterizes the automorphism groups of configuration spaces and discriminant varieties of algebraic curves, revealing their structure, invariants, and decompositions, and extends previous results with new proofs and bounds.

Contribution

It provides a detailed description of automorphisms of configuration spaces and discriminant hypersurfaces, including their Lie algebras and invariants, and improves previous results with broader assumptions and sharper bounds.

Findings

Automorphism group Aut$\, ext{Config}(X)$ is solvable.

Computed Lie algebra and Makar-Limanov invariant of configuration spaces.

Extended results to discriminant hypersurfaces and improved bounds.

Abstract

The configuration space $\mathcal{C}^n(X)$ of an algebraic curve $X$ is the algebraic variety consisting of all $n$-point subsets $Q\subset X$. We describe the automorphisms of $\mathcal{C}^n(\mathbb{C})$, deduce that the (infinite dimensional) group Aut$\,\mathcal{C}^n(\mathbb{C})$ is solvable, and obtain an analog of the Mostow decomposition in this group. The Lie algebra and the Makar-Limanov invariant of $\mathcal{C}^n(\mathbb{C})$ 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 \cite{Lin-Zaidenberg14}. 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 \cite{Lin-Zaidenberg14} and due to Zinde and to the first author. This allowed us to provide the optimal dimension bounds in our theorems.