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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 $L(Z)$?

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    $\begingroup$ I suspect you mean the set of effective cycles rationally equivalent to $[Z]$ in the second sentence. $\endgroup$ Oct 26, 2011 at 22:50
  • $\begingroup$ I know this is 5 years old, but the set you speak of at least appears in this paper for abelian varieties: projecteuclid.org/download/pdf_1/euclid.rmjm/1250126294 $\endgroup$
    – rfauffar
    Sep 30, 2016 at 1:18

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