  Along the lines of Hodge and Tate conjectures, Beilinson conjectured that in
the qth cohomology all the weight 2q Hodge cycles of a smooth complex variety
and all the weight 2q Tate cycles of a smooth variety over a finitely generated
field comes from the higher Chow groups. For product of curves and semiabelian
varieties, Beilinson-Hodge conjecture was shown in a previous paper by the
authors. Here both Beilinson-Hodge and Beilinson-Tate conjectures are shown to
be true for varieties dominated by product of curves. We also show that lower
weight Hodge cycles (resp. Tate cycles) are algebraic in these situations.
