  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.
