Proposition 9.6.21CAC94BB1C9D

Proof. In $X\times \mathbb P^ r$ consider the analytic set $Z$ defined by the equations of $X$ and by $f_ i\xi _ j=f_ j\xi _ i$ for all $i,j$; in the chart $\{ \xi _ i\neq 0\} $ these are the holomorphic equations $f_ j=f_ i\cdot (\xi _ j/\xi _ i)$ and $f_ k\cdot (\xi _ j/\xi _ i)=f_ j\cdot (\xi _ k/\xi _ i)$. Over a point $x\notin V(\mathfrak a)$, the only point of $Z$ is $(x,[f(x)])$, because $\xi $ must be proportional to the nonzero vector $f(x)$. Thus the graph is the part of $Z$ outside the analytic set $\pi ^{-1}(V(\mathfrak a))$, and by lemma 9.3.7 its closure is analytic; near each point it is the union of the branches of $Z$ not contained in $\pi ^{-1}(V(\mathfrak a))$. It is closed in $X\times \mathbb P^ r$, and $\mathbb P^ r$ is compact, so $\pi $ is proper. Over $X\setminus V(\mathfrak a)$ the blowup is the graph, and $\pi $ is an isomorphism there, with inverse $x\mapsto (x,[f(x)])$, which is holomorphic because on $\{ f_ i\neq 0\} $ the affine coordinates $f_ j/f_ i$ are holomorphic. The set $V(\mathfrak a)$ is nowhere dense in $X$, by corollary 9.3.6, and the graph is dense in the blowup; so $\pi $ is bimeromorphic.

On the chart $\{ \xi _ i\neq 0\} $ of the blowup, the relations $f_ j\circ \pi =(f_ i\circ \pi )\cdot (\xi _ j/\xi _ i)$ hold on the graph, hence everywhere, so the pulled-back ideal is generated by $f_ i\circ \pi $ there. This generator is not identically zero on any branch of the blowup at any point of the chart, because the graph points of the chart, at which $f_ i\circ \pi \neq 0$, are dense in every such branch. Hence it is a nonzerodivisor, and the pulled-back ideal is invertible.

Let $g\colon Y\to X$ be as in the statement. Near a point of $Y$, let $d$ be a nonzerodivisor generating $\mathfrak a\mathcal O_ Y$, and write $f_ i\circ g=db_ i$. Writing $d=\sum _ ic_ i\, (f_ i\circ g)=d\sum _ ic_ ib_ i$ and cancelling the nonzerodivisor $d$, we get $\sum _ ic_ ib_ i=1$; so the $b_ i$ have no common zero, and $y\mapsto (g(y),[b_0(y):\cdots :b_ r(y)])$ is a holomorphic map into $X\times \mathbb P^ r$. Where $d\neq 0$, some $f_ i(g(y))$ is nonzero, and this point is the graph point over $g(y)$; since $\{ d=0\} $ is nowhere dense, the whole image lies in the closure of the graph. Replacing $d$ by $ud$ for a unit $u$ replaces $b$ by $u^{-1}b$ and does not change the projective point, so these local lifts glue. A lift of $g$ is uniquely determined over $Y\setminus g^{-1}(V(\mathfrak a))=Y\setminus \{ d=0\} $, where $\pi $ is an isomorphism, hence everywhere by density.

Finally, the blowups defined by two families of generators both pull back $\mathfrak a$ to an invertible ideal. By the universal property, each maps to the other over $X$, and both compositions are the identity by uniqueness. These canonical isomorphisms also glue the local constructions on overlaps, compatibly on triple overlaps by the same uniqueness. The glued space is Hausdorff: points with different images in $X$ are separated by the inverse images of disjoint neighborhoods, and points with the same image lie in one of the local constructions, which are Hausdorff. Since $X$ has a countable basis, countably many of the open sets over which the local constructions are made cover $X$, and each local construction, an analytic subset of the product of a local model with $\mathbb P^ r$, has a countable basis; hence so has the glued space. Being locally compact and Hausdorff with a countable basis, it is paracompact. $\square$