In the context of complex manifolds, one can consider a blow-up along a complex submanifold. For a linear subspace of $\mathbb{C}^n$ there is a general procedure to perform such a blow-up: for a subspace $\mathbb{C}^m$ determined by the equations $$ z_1 = z_2 = \ldots = z_{n-m} = 0 \,, $$ we can introduce a projective space $\mathbb{CP}^{n-m-1}$ with coordinates $y_i$, and define the blown-up space $\tilde{X}$ by $$ \tilde{X} = \{(z,y)~|~z_i y_j = z_j y_i \,, ~i,j=1,\ldots,n-m\} \subset \mathbb{C}^n \times \mathbb{CP}^{n-m-1} \,, $$ with a blow-down map that simply projects onto the $\mathbb{C}^n$ factor. The equations ensure that the blow-down is an isomorphism away from the original subspace, and sends a $\mathbb{CP}^{n-m-1}$ to a point everywhere on the subspace.
For codimension $n-m=1$, the above procedure doesn't define a blow-up, since $\mathbb{CP}^{n-m-1}$ is just a point. My question is: Is there any similar construction for codimension one subspaces?