Modification of a complex space

Remarks

Example: the blow-up $f \colon X \to \C^2$ of $\C^2$ at the origin is a modification with $B = \set{0}$. It is proper and surjective, $f^{-1}(0)$ is a projective line, which is nowhere dense in the $2$-dimensional manifold $X$, and $f$ is biholomorphic off it.

Non-example: let $X$ be the disjoint union of $\C$ and a point $p$, and let $f \colon X \to \C$ be the identity on $\C$ and send $p$ to $0$. Then $f$ is proper and surjective, and with $B = \set{0}$ clause 3 holds. But $f^{-1}(B)$ contains the isolated point $p$, so it is not nowhere dense and clause 2 fails. No other choice of $B$ helps, since $f$ is not injective over $0$, so $p$ must lie over $B$.

The set $f^{-1}(B)$ is the exceptional set. Some authors call these maps proper modifications.