Let $X$ be an algebraic variety and let $Z\subset X$ be a subvariety. Let $[Z]$ be the class defined by $Z$ in the Chow group. Let $L(Z)$ be set of effective algebraic cycles on $X$ linearly equivalent to $[Z]$. When $Z$ is an effective divisor, this set is a linear system, and so has the structure of a projective space.
Is there any reasonable geometric structure on $L(Z)$ in general?
If so, has this problem been studied before? In particular, can one give estimates for the dimension of $L(Z)$?