Definition 9.6.1757D038DF549

Definition 9.6.1.757D038DF549 Let $X$ be a complex space and let $\mathfrak a\subset \mathcal O_ X$ be a coherent ideal sheaf that is not identically zero on any branch of $(X,x)$, for every $x\in X$. On an open set where $\mathfrak a$ is generated by $f_0,\ldots ,f_ r$, the blowup $\operatorname {Bl}_{\mathfrak a}X$ is the closure, with its reduced structure, of the graph

Here $\mathbb P^ r$ is the space of nonzero vectors in $\mathbb C^{r+1}$ modulo multiplication by nonzero scalars, with the charts $\{ \xi _ i\neq 0\} $ and the affine coordinates $\xi _ j/\xi _ i$. It is compact, because every point has a representative on the compact unit sphere and the quotient map is continuous. The projection to $X$ is denoted by $\pi $.