Example 9.4.1060A14C7A1845

Example 9.4.10.60A14C7A1845 The curve $\{ zw=0\} \subset \mathbb C^2$ has two branches at the origin, the coordinate axes, whose germs are irreducible, being germs of submanifolds (lemma 9.3.12). The disjoint union of the two axes, mapped to the curve by the inclusions, is a manifold, hence normal, and the map is finite, bimeromorphic and an isomorphism over the regular locus $\{ zw=0\} \setminus \{ 0\} $; by the uniqueness in theorem 9.4.6 it is the normalization. Its fiber over the origin has two points, and the curve is not unibranch.

The cusp $C=\{ w^2=z^3\} $ is normalized by $\gamma (t)=(t^2,t^3)$. Indeed, $\gamma $ is a holomorphic bijection of $\mathbb C$ onto $C$: for $(z,w)\in C$ with $z\neq 0$, the equations $t^2=z$, $t^3=w$ force $t=w/z$, and $t=w/z$ satisfies them, since $(w/z)^2=z$ and $(w/z)^3=(w/z)z=w$; and $\gamma (t)=0$ only for $t=0$. The points of $C\setminus \{ 0\} $ are regular: at such a point $z\neq 0$, so the map $(z,w)\mapsto (w^2-z^3,w)$ has the nonzero Jacobian determinant $-3z^2$ there, and by proposition 5.2.4(1) the functions $w^2-z^3$ and $w$ form holomorphic coordinates near such a point, in which $C$ is a coordinate line; so $C$ is a submanifold there. On $C\setminus \{ 0\} $ the inverse of $\gamma $ is the holomorphic function $t=w/z$. This function satisfies $t^2=z$, so it tends to $0$ at the origin; hence $\gamma $ is a homeomorphism.

The function $t$ is not the restriction of a holomorphic function $F(z,w)$ near the origin, because $F(t^2,t^3)$ has no term of degree one in $t$. Hence the origin is a singular point of $C$. Otherwise $t$, which is bounded near the origin, would extend holomorphically across it on the manifold $C$, by lemma 9.2.9, and would therefore be the restriction of a holomorphic function of $(z,w)$: in coordinates in which a submanifold is a coordinate plane, a holomorphic function on it extends independently of the other coordinates. So $t$ is weakly holomorphic on $C$, and integral, since $t^2=z$, but it is not holomorphic at the origin, so $C$ is not normal there. Thus $\gamma $ is a finite bimeromorphic map from a manifold that is an isomorphism over $C_{\mathrm{reg}}=C\setminus \{ 0\} $, and by the uniqueness in theorem 9.4.6 it is the normalization. The cusp is therefore unibranch but not normal, and it distinguishes unibranch spaces from normal spaces.