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.