2 fixes

I might be missing something about question 3. Here's a simple construction.:

Consider a projective space $P$ of dimension $\text{dim}\, P_1 + \text{dim}\, P_2$ that contains both $P_1$ and $P_2$ in general position. Then each point of $P-P_1-P_2\$ lies on exactly one line connecting $x$ from $P_1$ with $y\in P_2$. Is this the kind of join you're looking for?

About question 2, I have a simpler thing that isn't clear to me (now posted as a questionfirst):

is a complex algebraic variety which is topologically contractible necessarily an affine line? If not, is it necessarily affine?

1

I might be missing something about question 3. Here's a simple construction.

Consider a projective space $P$ of dimension $\text{dim}\, P_1 + \text{dim}\, P_2$ that contains both $P_1$ and $P_2$ in general position. Then each point of $P-P_1-P_2\$ lies on exactly one line connecting $x$ from $P_1$ with $y\in P_2$. Is this the kind of join you're looking for?

About 2, I have a simpler question first: is a complex algebraic variety which is topologically contractible necessarily an affine line? If not, is it necessarily affine?