Local smoothing and nodal fiber of plumbing

Fix $q\mathrm{\ne }0$. On the fiber $zw=q$, neither $z$ nor $w$ vanishes. The map $z\mapsto (z,q\mathrm{/}z)$ is holomorphic on the annulus $\mathrm{\mid }q\mathrm{\mid }\mathrm{/}s<\mathrm{\mid }z\mathrm{\mid }<r$, has holomorphic inverse given by projection to $z$, and identifies it with the corresponding $w$-annulus by $w=q\mathrm{/}z$. Thus the sewn chart transition is biholomorphic, and every point in this local fiber has a one-dimensional complex chart.

At $q=0$, the equation is $zw=0$, so its point set is the union of the coordinate axes $\{z=0\}\cup \{w=0\}$. Its analytic local ring is $\mathbb{C}\{z,w\}\mathrm{/}(zw)$; a convergent power series divisible by both $z$ and $w$ has every monomial divisible by $zw$, so $(z)\cap (w)=(zw)$ and the ring is reduced. The classes of $z$ and $w$ are independent in the cotangent space because the relation has no linear term. Hence the tangent space at the origin is two-dimensional while each branch has complex dimension one. This is the ordinary node in the local analytic convention of the Statement, and it is not a smooth Riemann-surface point.

For disjoint plumbing pairs, each local equation uses only its own coordinate pair $(z_j,w_j)$ and parameter $q_j$. The product charts therefore have independent biholomorphic transitions for every $q_j\mathrm{\ne }0$, while setting all $q_j=0$ gives the product of the independent equations $z_j{w}_j=0$. The smoothness and node conclusions of steps 1.1–1.2 apply in each disjoint neighborhood, with no cross-identification between pairs.