Let $Z$ be a projective variety embedded into $\mathbb P^n$. Then we can define an affine cone over $Z$ as the inverse image of $Z$ under canonical map $\mathbb A^{n+1}\setminus0 \to \mathbb P^n$. I have to questions about this construction:

- Does a cone over the given projective variety $Z$ depend on an embedding of $Z$ into $\mathbb P^n$?
- Suppose we are given an affine variety $X$. Can it be an affine cone over two nonisomorphic projective varieties?

By the same method we can define a weighted affine cone for a variety $Z$ embedded into weighted projective space $\mathbb P(k_0,..k_n)$. Are the same properties true for weighted projective cone?