Let $X$ be an connected surface which set-theoretically is the union of $\mathbb{P}^2-pt$ and $\mathbb{P}^1$. Suppose there is a birational morphism $\pi:X\to\mathbb{P}^2$ such that all of $\mathbb{P}^1$ above is mapped to the point. Does this imply that $X$ is the blow-up of $\mathbb{P}^2$ at that point?
If we know in addition that $X$ is smooth, then it is classical that the existence of $\pi$ implies $X$ is the blow-up. But is it possible for some singular surface $X$ to have the above property?
Thanks for the help!