Let $S$ be a smooth projective surface. $Bl_p(S)$ be the blow up of $S$ at a point $P$. Let $Y$ be a variety. If we have: $f:Bl_p(S)\to Y$, $g:Y\to S$ such taht $f\circ g$ is the natural projection, and $f$ is a bijection, can we conclude that $f$ is an isomorphism?
Thank you!
Addition related to comment below: I was aware of the excellent linked post. However, one key ingredient missing here is the normality of $Y$. I am wondering if the additional information in my situation can make up for the loss of normality here.