"Let $X, Y \subset \mathbb{C}^4$ be varieties defined by $$ X := \{ (t,t^2,t^3,0) \,|\, t \in \mathbb{C} \}, \quad Y := \{ (0, u, 0, 1) \,|\, u \in \mathbb{C} \}. $$
The join variety of $X$ and $Y$ is the set $$ J(X,Y) := \bigcup_{P\in X, Q\in Y} \overline{PQ} \subset \mathbb{C}^4, $$
where $\overline{PQ}$ is the line through $P$ and $Q$. Describe the set $J(X,Y)$.
Is $J(X,Y)$ an affine (or quasi-affine) variety?"
This is a problem in the chapter 1 of Klaus Hulek's algbraic geometry. I can somehow visualize what the join variety is in my head, but can't figure out at all if this is some variety.
Could anyone give me some answers?