Let $X$ be a closed subscheme of $\mathbb{A^n}$ (over a basefield) defined by an ideal $I$ and consider the immersion $\mathbb{A^n}\to \mathbb{P^n}$, $(x_1,\ldots, x_n)\mapsto [x_1,\ldots,x_n,1]$. One may consider the projective variety $\bar X$ in $\mathbb{P^n}$ given by the homogenized ideal $\bar I$. This ideal consists of the homogenized elements of $I$, so for example if $f=x_1^2+x_2+1$ is in $I$ then $\bar f=x_1^2+x_2x_{n+1}+x_{n+1}^2$ is in $\bar I$. Then $\bar X$ is the projective closure of the image of $X$ under $\mathbb{A^n}\to \mathbb{P^n}$, right? I wonder if smoothness is inherited.
If $X$ is a smooth affine scheme over the base field, is $\bar X$ smooth, too?
Best Answer
No, smoothness of $X$ is not inherited by $\bar X$.
Take two parallel lines in $\mathbb A^2_k$. Their union $X\subset \mathbb A^2_k$ is a smooth scheme but the lines meet at a singular point of $\bar X $ at infinity, so that $\bar X \subset \mathbb P^2_k$ is no longer smooth.
Renaissance artists, who invented perspective, have given us beautiful paintings of these schemes.