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:


1. Does a cone over the given projective variety $Z$ depend on an embedding of $Z$ into $\mathbb P^n$?
2. 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? 



Uniqueness of a (weighted) affine cone
Abotu the first question: yes, it does. Already for $\mathbb{P}^n$ embedded with $\mathcal{O}(d)$, the cone changes with $d$.


About the second, you might view a cone as a variety with a $\mathbb{G}\_m$ and just one fixed point. On a fixed variety, you might have two different actions giving different cone structures.


