Question

Let $k$ be an algebraically closed field, and $V$ a normal (irreducible) affine variety over $k$. Does there necessarily exist a closed immersion $V \hookrightarrow \mathbb{A}^n$ of $V$ into affine space such that the closure of $V$ in projective space $\mathbb{P}^n$ is normal?

Answer 1

Yes:

Take a closure $\bar X$ of $X$ in some projective space. We can write $\bar X= X\cup D$, with $D$ ample. Normalize $\bar X$ to get new projective variety $\pi:\tilde X\to \bar X$. The preimage $\pi^{-1}D$ is ample with complement $X$ because $\pi$ is finite. So $\tilde X$ can be re-embedded in another projective space $\mathbb{P}^N$ so that $\pi^{-1}D$ is set theoretically the intersection of $\tilde X$ with a hyperplane $H$. Under the embedding of $X\subset \mathbb{A}^N=\mathbb{P}^N-H$, the closure is $\tilde X$ which is normal.