Local parametrization theorem

Remarks

The integer $d$ is the dimension of the germ $A_0$, and the coordinates can be taken generic: a dense open set of linear coordinate systems works. The proof uses the ideal $\mathfrak{p}$ of $A_0$, which is prime, and Noether normalization in the analytic setting. After a generic change of coordinates, the Weierstrass preparation theorem makes $\mathcal{O}_{n,0}/\mathfrak{p}$ a finite extension of $\mathcal{O}_{d,0}$, the germs in $z'$ alone. Here $d$ is the largest number of variables for which $\mathcal{O}_{d,0} \to \mathcal{O}_{n,0}/\mathfrak{p}$ is injective. A primitive element of the fraction field extension and its discriminant give the set $\Delta$ and the number $k$ of sheets. Example: for the cusp $A = \set{z_1^2 = z_2^3}$ in $\C^2$, take $z' = z_2$ and $\pi(z_1, z_2) = z_2$. Over $z_2 \ne 0$ the fibre consists of the two square roots of $z_2^3$, so $\Delta = \set{0}$ and $k = 2$; the set $A \setminus \set{0}$ is the image of the punctured disc under $t \mapsto (t^3, t^2)$, hence connected. Irreducibility is needed for the connectedness in part 3. For $A = \set{z_1 z_2 = 0}$ and any linear projection to a line with finite fibres on $A$, the complement of $\pi^{-1}(\Delta)$ has a component in each axis.

Going deeper

For a student

The local parametrization is the basic structure theorem of local analytic geometry. It gives the dimension of an analytic set and shows that the regular points are dense. It also yields the Rückert Nullstellensatz, and it is the analytic counterpart of Noether normalization for affine varieties. The number $k$ of sheets for generic coordinates is the multiplicity of the germ at $0$.