Discriminant and branch set of a fixed Weierstrass projection

Fix $n\ge 1$ and a reduced nonzero nonunit germ $f\in {\mathcal{O}}_{{\mathbb{C}}^n,p}$. Center at $p$ and fix the data supplied by Finite local projection of a reduced hypersurface germ: an invertible complex-linear map $T$, the affine coordinate map $\mathrm{\Phi }(z)=p+Tz$, a preparation

$f\circ \mathrm{\Phi }=uW(z^{\mathrm{\prime }},t),$

and its chosen product representative $V\times D$. Here $W$ is monic of degree $d\ge 1$ in $t$, with coefficients in ${\mathcal{O}}_{n-1,0}$, and $u$ is nonvanishing on the representative. Use the product neighbourhood of the finite-projection theorem, on which every slice has all its $d$ roots, counted with multiplicity, inside $D$ and none on $\mathrm{\partial }D$; mere agreement of zero sets on an arbitrary product does not suffice. Put $X_W:=Z(W)\cap (V\times D)$ and let $\pi :X_W\to V$, $(z^{\mathrm{\prime }},t)\mapsto z^{\mathrm{\prime }}$, be the restricted coordinate projection. This chosen projection is proper and surjective, and is a $d$-sheeted holomorphic covering off the discriminant.

Discriminant. The discriminant of the fixed prepared equation is the base germ

$D_W(z^{\mathrm{\prime }}):={\mathrm{Disc}}_t(W)\in {\mathcal{O}}_{n-1,0},$

the coefficient expression of The discriminant of a monic polynomial as the coefficient expression of ${\mathrm{\Delta }}_n^2$ applied to the monic polynomial $W(z^{\mathrm{\prime }},\cdot )$ over ${\mathcal{O}}_{n-1,0}$.

Branch set. The branch set of the projection $\pi$ is

$B_{\pi }:=\{z^{\mathrm{\prime }}\in V:D_W(z^{\mathrm{\prime }})=0\},$

the zero set of the discriminant germ inside the chosen base neighbourhood.

The definition is well posed for the fixed equation and projection:

1. The Weierstrass polynomial is unique for that fixed linear projection, so it does not change if the original germ is multiplied by a unit, or if another preparation of the same regular germ is used (Uniqueness in Weierstrass preparation).
2. The discriminant is not identically zero: $D_W\mathrm{\ne }0$ as a germ (Reduced preparation has nonzero discriminant).
3. For $z_0^{\mathrm{\prime }}\in V$ the value $D_W(z_0^{\mathrm{\prime }})=\mathrm{Disc}(W(z_0^{\mathrm{\prime }},\cdot ))$ vanishes exactly when the slice has a repeated root (The discriminant is ${\prod }_{i<j}({\alpha }_i-{\alpha }_j{)}^2$ and vanishes exactly when a monic polynomial has a repeated root). Since $W$ is monic of degree $d$, this happens exactly when the fibre ${\pi }^{-1}(z_0^{\mathrm{\prime }})$ is not a set of $d$ distinct simple roots, that is, exactly where the $d$ unramified local sheets supplied by the finite-projection theorem fail to exist. Thus $B_{\pi }$ is the base locus of the branching of $\pi$.

The branch set belongs to the chosen projection: it is defined after fixing the linear coordinate change $T$ and the base neighbourhood, and it need not equal the image under $\pi$ of the singular locus of the hypersurface. The companion examples page exhibits the smooth curve $y^2=x$, whose chosen projection $(x,y)\mapsto x$ is branched at $x=0$ although the curve has no singular point; the singular locus itself is defined independently of any projection on this page.