Several complex variables and analytic spaces Definition

Isolated hypersurface singularity

Builds on Holomorphic function of several variables Germ Critical point

Definition

Let $n \ge 0$ , let $U$ be an open neighbourhood of $0$ in $\C^{n+1}$ and let $f \colon U \to \C$ be a holomorphic function with $f(0) = 0$ . The germ $f$ has an isolated singularity at $0$ if:

$\frac{\partial f}{\partial z_j}(0) = 0$ for $j = 0, \dots, n$ , so $0$ is a critical point of $f$ ;
there is a neighbourhood $U' \subseteq U$ of $0$ in which $0$ is the only point where all the partial derivatives $\partial f / \partial z_j$ vanish.

The germ at $0$ of the hypersurface $V = f^{-1}(0)$ is then called an isolated hypersurface singularity.

Remarks

Example: $f(z) = z_0^2 + z_1^2 + \cdots + z_n^2$ . Its gradient is $(2z_0, \dots, 2z_n)$ , which vanishes only at $0$ , so both clauses hold.

Non-example: $f(z_0, z_1, z_2) = z_0 z_1$ on $\C^3$ . Its gradient is $(z_1, z_0, 0)$ , which vanishes on the whole $z_2$ -axis, so clause 2 fails.

Let $J$ be the ideal of the ring $\C\{z_0, \dots, z_n\}$ of convergent power series generated by the partial derivatives of $f$ . By the Rückert Nullstellensatz, clause 2 holds if and only if $\C\{z_0, \dots, z_n\} / J$ is finite-dimensional over $\C$ . Its dimension is the Milnor number of $f$ . If $f$ has no repeated factor, the points near $0$ where $V$ is singular are the points of $V$ where the gradient of $f$ vanishes.

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Critical point concept
Germ concept
Holomorphic function of several variables concept
Isolated hypersurface singularity definition

Tag	`4KAWK0FD4V`
URI	`m2://4KAWK0FD4V`
Kind	definition
Writing	W0 stub
MSC	32S25, 32S05, 14B05
Defines	Isolated hypersurface singularity
Uses	Holomorphic function of several variables, Germ, Critical point

Isolated hypersurface singularity

Definition

Remarks

Reading path

Further reading

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