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I am working through Karen Smith's Invitation to Algebraic Geometry and one of the problems is: Prove that every projective variety is the compactification of an affine variety in the Zariski topology and Euclidean topology. By compactification, we mean a compact extension such that the base set is open and dense in the compact extension.

I have already shown that projective varieties are compact. Now I just need to show that they are compactifications of affine varieties. So far, I have shown the following: Let $V \subset \mathbb{P}^n$ be a projective variety such that $V \not\subset \mathbb{V}(x_0)$ and suppose without loss of generality that $V \cap \mathbb{V}(x_0) \ne \emptyset$. Then, since $V \cap \mathbb{V}(x_0)$ is closed, we know that $V \setminus (V \cap \mathbb{V}(x_0)) \subset \mathbb{P}^n \setminus \mathbb{V}(x_0) \cong \mathbb{A}^n$ is an affine variety that is open in the subspace topology on $V$. Since $V$ is compact, we only need to show that $V \setminus (V \cap \mathbb{V}(x_0))$ is dense in $V$.

I am a little unsure about how to proceed from here and wanted to know if my work so far has been correct.

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  • $\begingroup$ Yes youre right @peterag $\endgroup$ – Yunus Syed Feb 5 at 22:00

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