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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?

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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.

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    $\begingroup$ To expand Giulio's answer: one of the property of the affine cone that does change under different embedding is its deformation theory. To give an example, assume that $X$ is a smooth projective (projectively normal as well) variety. It is not difficult to show that the affine cone $C(v_d(X))$ of a sufficiently high degree Veronese embedding of $X$ will have only conical deformations, whereas in general the deformation theory of $C_X$ will be richer (it can contains, for example, smoothings). $\endgroup$
    – Enrico
    Commented Mar 14, 2016 at 0:55

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