The \(\text{SL}_ 2(\mathbb C)\)- and \(\text{PSL}_ 2(\mathbb C)\)-character varieties of the fundamental groups of hyperbolic \(3\)-manifolds have played an important role in \(3\)-dimensional topology. In this article, the authors give a careful and fully referenced development of the algebraic theory of \(\text{PSL}_ 2(\mathbb C)\)-representation varieties and their associated character varieties. As an application, they show that for every \(n\), there is a hyperbolic punctured-torus bundle over \(S^1\) whose \(\text{PSL}_ 2(\mathbb C)\)-character variety has at least \(n\) irreducible one-dimensional components whose characters do not lift to \(\text{SL}_ 2(\mathbb C)\). Additional results about singular sets of character varieties are obtained, in particular, the singular set is computed for the \(\text{PSL}_ 2(\mathbb C)\)-character variety of a free group \(F_n\), \(n\geq 3\).