For the sake of this question for simplicity you can assume $X$ is a hypersurface in the complex projective space (not necessarily smooth). Let $C_{r,d}(X)$ be the quasi-projective Chow variety corresponding to degree $d$ and $r$ dimensional irreducible cycles on $X$. This is different from the projective Chow variety that contain all cycles and is denoted by $\overline{C_{r,d}(X)}$.
- The question is about the primes that appear as torsion in the Borel-Moore homology of $C_{r,d}(X)$. Let $S_n$ be the set of primes that appear as a torsion in $H^{B.M.}_{n}(C_{r,d}(X))$ for infinite number of $d$'s. What does the set $S_n$ look like? can it contain all primes?
