I'm trying to get an intuition for what things look like in projective coordinates. There are two curves that I have to work a problem with, but I'm not sure how to visualize them. They are
$V(u^2 X + v^2 Y + uvZ) \subset \mathbb{P}^2 \times \mathbb{P}^1 $
$V(u^2 X^2 + v^2 Y^2 + uvZ^2) \subset \mathbb{P}^2 \times \mathbb{P}^1$
where $[u : v]$ are the projective coordinates of $\mathbb{P}^1$ and $[X : Y : Z]$ are the projective coordinates of $\mathbb{P}^2$. Is there any way to describe these objects geometrically?