Proposition 9.6.3865FE5EC97E6

Proof. Let $(z,w)$ be coordinates on an open set $U$ as in the first part of the statement; such coordinates exist near every point of $C$. On $U$ the ideal $\mathcal I_ C$ is generated by $w_1,\ldots ,w_ r$, by lemma 9.3.12. Let $Z\subset U\times \mathbb P^{r-1}$ be the analytic set defined by the equations $w_ j\xi _ i=w_ i\xi _ j$. Over a point of $U\setminus C$, its only point is $((z,w),[w])$, because $\xi $ must be proportional to the nonzero vector $w$; so the part of $Z$ over $U\setminus C$ is the graph of definition 9.6.1, whose closure is the blowup of $U$. In the chart $\{ \xi _ i\neq 0\} $, the equations $w_ j\xi _ i=w_ i\xi _ j$ say that $w_ j=w_ iv_ j$ with $v_ j=\xi _ j/\xi _ i$, $j\neq i$; hence this part of $Z$ is the graph of the map $(z,t,v)\mapsto (z,w)$ with $w_ i=t$ and $w_ j=tv_ j$, $j\neq i$. It is a manifold with coordinates $(z,t,v)$, in which the graph part is $\{ t\neq 0\} $, a dense subset. Hence these charts cover the blowup, $M'$ is a manifold, and $\mathcal I_ C\mathcal O_{M'}$ is generated by $t$ in each chart. Thus $E=\{ t=0\} $ is a smooth hypersurface.

With $w_ i$ listed first among the coordinates $w$, the Jacobian matrix of (9.6.4) is block lower triangular with diagonal blocks $I$, $1$ and $tI_{r-1}$, because $\partial w_ i/\partial t=1$, $\partial w_ j/\partial t=v_ j$ and $\partial w_ j/\partial v_ l=t\delta _{jl}$ for $j,l\neq i$, while $z$ does not depend on $(t,v)$ and $w$ does not depend on $z$; so the Jacobian determinant of $\pi $ is $\pm t^{r-1}$. The fiber over a point $(z,0)$ is obtained by gluing the coordinate charts $v\in \mathbb C^{r-1}$, that is, it is $\mathbb P^{r-1}$. If $r=1$, there is one chart, and the coordinates and the map are unchanged.

For the second part, choose coordinates $(z,w)$ near a point of $C$ with $W=\{ w_1=\cdots =w_ s=0\} $ and $C=\{ w_1=\cdots =w_ s=z_1=\cdots =z_ k=0\} $, where $k\geq 1$ is the codimension of $C$ in $W$. Such coordinates exist: take coordinates $(x,w)$ with $W=\{ w=0\} $, then holomorphic coordinates $z=\phi (x)$ on $W$ near the point in which $C$ is a coordinate plane, and use $(\phi (x),w)$, whose Jacobian matrix is block triangular and invertible, so that these are holomorphic coordinates near the point by proposition 5.2.4(1). The center is now generated by $z_1,\ldots ,z_ k,w_1,\ldots ,w_ s$. In a chart whose pivot is $z_ i$, we have $z_ i=t$, $z_ l=tu_ l$ for $l\neq i$, $l\leq k$, and $w_ j=tv_ j$; here $\pi ^{-1}(W\setminus C)=\{ v=0,\ t\neq 0\} $, so $W'=\{ v=0\} $ in this chart, and

In a chart whose pivot is $w_ i$, we have $w_ i=t$, so $\pi ^{-1}(W)$ lies in $\{ t=0\} =E$. Moreover, $W'$ does not meet this chart, because the points of $\pi ^{-1}(W\setminus C)$, and therefore those of its closure, have projective coordinates $\xi $ whose components corresponding to $w_1,\ldots ,w_ s$ vanish. The pulled-back ideal contains $t$ and is therefore $(t)=\mathcal O_{M'}(-E)$, which proves (9.6.5) in this chart as well. In the charts of the first kind, $W'$ has the coordinates $(t,u)$ together with the remaining $z$-coordinates, and $\pi |_{W'}$ is given by the formulas of the blowup of $W$ along $C$; so $W'=\operatorname {Bl}_ CW$. Finally, if $C$ is disjoint from $W$, then $\pi $ is an isomorphism over the neighborhood $M\setminus C$ of $W$. $\square$