Infinitely generated Lawson homology groups on some rational projective varieties

We construct rational projective 4-dimensional varieties with the property that certain Lawson homology groups tensored with ${ \mathbb{Q}}$ are infinite dimensional ${\mathbb{Q}}$-vector spaces. More generally, for each pair of integers $p$ and $k$, with $k\geq 0$, $p>0$, we find a projective variety $Y$ such that $L_pH_{2p+k}(Y)$ is infinitely generated.

We also construct two singular rational projective 3-dimensional varieties $Y$ and $Y'$ with the same homeomorphism type but different Lawson homology groups; specifically, $L_1H_3(Y)$ is not isomorphic to $L_1H_3(Y')$ even up to torsion.