Proving that Projective Varieties are Compactifications of Affine Varieties

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.

• Yes youre right @peterag – Yunus Syed Feb 5 at 22:00