Blowing up of affine space

Question

I am learning blowing up from An Invitation to Algebraic geometry (Karen E. Smith). In the chapter 7 (103- page) written that blowing up of affine space $\mathbb{A}^n$ along point $p$ is not affine variety. How I can prove it? I am trying to use the fact that : only global regular functions on a projective variety are constant functions. Could anyone help me to prove it?

Answer 1

Let $X$ be the blow-up of $\mathbb{A}^n$ at a point $p$. Then $X$ admits a closed immersion $\mathbb{P}^{n-1}\to X$ with image $\varphi^{-1}(p)$. So $X$ can't be affine, because otherwise $\mathbb{P}^{n-1}$ would be affine as well, which it isn't for $n\geq 2$. But note that $\mathbb{P}^0$ is affine, and blowing up $\mathbb{A}^1$ at a point does nothing, so the blow-up remains affine.

Let us examine this more in detail: suppose for simplicity that $p=O$ is the origin. The blow-up $X$ of $\mathbb{A}^n$ at $O$ may be defined as the closed subset of $\mathbb{A}^n_{x_1,\ldots,x_n}\times\mathbb{P}^{n-1}_{y_1,\ldots,y_n}$ defined by the equations $x_iy_j=x_jy_i$ where $i,j$ run through $1,\ldots,n$. More formally, note that we can describe the closed subsets of $\mathbb{A}^n_{x_1,\ldots,x_n}\times\mathbb{P}^{n-1}_{y_1,\ldots,y_n}$ by ideals of $k[x_1,\ldots,x_n][y_1,\ldots,y_n]$ which are generated by polynomials homogeneous in $y_1,\ldots,y_n$ (e.g. the ideal $(y_1+x_1y_2)$). So with this description, $X$ is cut out by the ideal $I=(x_iy_j-x_jy_i\mid i,j\in\{1,\ldots,n\})$. Now $I$ is certainly contained in $J=(x_1,\ldots,x_n)$, and the set cut out by $J$ is precisely $\{O\}\times\mathbb{P}^{n-1}_{y_1,\ldots,y_n}\cong\mathbb{P}^{n-1}$. Hence $X$ contains the closed subset $\{O\}\times\mathbb{P}^{n-1}_{y_1,\ldots,y_n}$, so it can't be affine.