A function f: ($\mathbb{C}^n$,0) $\to$ ($\mathbb{C}$,0) is considered a hypersurface singularity if the point $(0,0,\dots,0)$ is the only point in the ideal $\langle \frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \dots, \frac{\partial f}{\partial x_n} \rangle$. The hessian matrix $(\frac{\partial^2 f}{\partial x_i x_j})$ gives rise to an invariant called the corank which is n - rank of the hessian. In Arnold's book (Singularities of Differentiable Maps V.1) there is a very algorithmic way of classifying singularities with corank 3 and less. For corank 4 and higher it's stated that they belong to a family denoted $\mathbf{O}$.

My question is what is known about corank 4 singularities? Are there any references which discuss how to classify these singularities?


I'm not sure that the definition at the beginning is correct. Let's assume that we have a holomorphic function germ $f : (\mathbb{C}^n,0) \to (\mathbb{C},0)$. Let $\mathscr{O}_n$ denote the local ring of holomorphic function germs $(\mathbb{C}^n,0) \to (\mathbb{C},0).$ Furthermore, let $J_f$ denote the Jacobian ideal generated by the partial derivatives of $f$, i.e. $J_f := (\partial f/\partial z_1,\ldots,\partial f/\partial z_n).$ The local algerba of $f$, denoted $A_f$, is given by

$A_f := \mathscr{O}_n/J_f .$

The local algebra is a complex vector space. The dimension is finite if and only if $f$ has an isolated critical point at $0 \in \mathbb{C}^n.$ The dimension $\dim_{\mathbb{C}}\left(A_f\right)$ is called the Milnor Number of $f$ at $0 \in \mathbb{C}^n$ and is equal to the absolute value of the Poincaré-Hopf index of the gradient vector field $\nabla f$ at $0 \in \mathbb{C}^n.$

The set of $z \in \mathbb{C}^n$ such that $f(z) = 0$ has a hypersurface singularity if the Milnor Number, $\mu > 0.$ The corank one singularities are given by the $A_{\mu}$ series where $\mu \ge 1$ ($\mu = 1$ gives a Morse type singularity, i.e. a non-degenerate critical point). The corank two singularities are given by the $D_{\mu}$ series where $\mu \ge 4.$ There are also three $E_{\mu}$ singularity types where $\mu = 6, 7, 8.$ The $A_{\mu}$, $D_{\mu}$, $E_6$, $E_7$ and $E_8$ are the only simple singularities.

That means that under the action of $\mathscr{A}$-equivalence (diffeomorphic changes of coordinate in the source and the target), any sufficiently small neighbourhood of a function germ representative of a simple singularity intersects only a finite number of other orbits. It's related to the modality of Lie group actions.

The book only deals with corank $\le 3$. Anything higher and the classification becomes too wild (and there aren't really any applications because these functions would be horribly non-generic). To get some idea, the set of function in $n$ variables with corank $k \le n$ has codimension $\frac{1}{2}k(k+1)$ in the space of function germs with critical points at $0.$ That means that the set of functions with corank four has codimension 10. You would need a generic 10-parameter family of functions to except a single member to have a corank four singularity.

There is no series of singualrities called O. That notation is used in adjacency diagrams to represent some variable series. After the simple singularities (the countable families) there a uni-modular, bi-modular, ..., even functionally modular families. A modular family means there are a continuum of non-equivalent members of the family. Take a look at pages 242 - 257. There's a whole zoo of singularity types, but no O's .

The bottom line is that, as far as I know, if you can classify the corank four singularities then you'd write a book in the process and you'd be very famous. (You'd have done what Arnol'd and his school couldn't or wouldn't do.) Even for small corank you have moduli. So you'd have families with uncountably many non-$\mathscr{A}$-equivalent members, i.e. functions. The only hope would be to use topological equivalence in order to reduce the moduli, but then you lose the point.


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