# If a partial normalisation of an affine variety is affine?

Let $X$ be an affine variety. Let $x$ be an isolated singular point of $X$. Let $U$ is an affine neighbourhood of $x$ such that $U\setminus \{x\}$ is smooth. Let $\pi:\tilde{U}\to U$ be the normalisation of $U$. We have $\tilde{U}\setminus\pi^{-1}(x)\simeq U\setminus \{x\}.$

We can construct a new variety $\tilde{X}$ by gluing $X\setminus \{x\}$ and $\tilde{U}$ by identifying $U\setminus\{x\}$ and $\tilde{U}\setminus\pi^{-1}(x)$. Is $\tilde{X}$ affine?

Thanks.

• For every finite morphism whose target is affine, also the domain is affine. – Jason Starr Mar 23 '18 at 23:53