  Given an action of the one-dimensional torus on a projective variety, the
associated Chow quotient arises as a natural parameter space of invariant
$1$-cycles, which dominates the GIT quotients of the variety. In this paper we
explore the relation between the Chow and the GIT quotients of a variety,
showing how to construct explicitly the former upon the latter via successive
blowups under suitable assumptions. We also discuss conditions for the
smoothness of the Chow quotient, and present some examples in which it is
singular.
