This is a report on joint work with A. Rosenschon. We show that on such a
3-fold, for all but a finite number of primes  $l$, the CHow group
of curves with mod $l$ coefficients is not finitely generated. This is done
in two steps: first we use a varaiant of the technique of  Bloch and Esnault to
show that the Ceresa cycle is not $l$-divisible for almost all $l$. Then we
use modular correspondences, following Nori, to show inifnite generation.