Let $V,W$ be two $4$-dimensional vector space over a field, $G(2,V),G(2,W)$ the Grassmanninas of planes in $V$ and $W$, $S_V$, $Q_V$ the tautological subbundle and quotient bundle of $G(2,V)$, and $S_W$, $Q_W$ the tautological subbundle and quotient bundle of $G(2,W)$.
In a research paper I read that $\mathbb{P}(Hom(S_W,Q_V))$ and $\mathbb{P}(Hom(S_V,Q_W))$ are projective bundles over $G(2,V)\times G(2,W)$.
What is the exact meaning of this? I am confused because we are taking Hom between bundles on different varieties. What is the rank of $\mathbb{P}(Hom(S_W,Q_V))$ and $\mathbb{P}(Hom(S_V,Q_W))$?