Is regular birational image of affine subset affine?

Question

Let $X$ and $Y$ be projective varieties and $φ\colon X \to Y$ a regular birational map. If $U⊆X$ is an affine subset of $X$, is $φ(U)$ affine?

This is related to my previous question.

Answer 1

No.

The simplest example is where $\varphi: X \rightarrow Y$ is the blowup of a point in $\mathbf P^2$. Then $X$ contains an affine open subset $U \cong \mathbf A^2$ and $Y$ contains an affine open subset $V \cong \mathbf A^2$ such that the map $\varphi: U \rightarrow V $ is given by the formula $(x,y) \mapsto (x,xy)$. So $\varphi(U) \subset V \cong \mathbf A^2$ is the union of the open set $\{x \neq 0\}$ and the point $(0,0)$. This is not affine.