I'm reading Singularity Theory (one of the authors is Arnold). I am a bit confused on the concept of modality. Is there an easy geometric description of it? The book has a relation between codimension, modality and multiplicity (Milnor number) but it confuses me a bit. Maybe as a person with algebraic geometry background I get confused with some of the terms.
Going back to modality, the authors first define the modality of a point on a manifold: To define this they need to always speak of some action of a Lie group on the manifold. And then they define the modality to be the least number $m$ such that a sufficiently small neighbourhood at the point is covered by finite number of "$m$-parameter" families of the orbits. This already is confusing to me. After that they define the modality of a function germ $f:(\mathbb C^n,0)\to(\mathbb C^n,0)$ in terms of its jets. The definition is not clear to me: They write it is the "modality of any its $k$-jets (which is to me basically an equivalence class of function with the same Taylor polynomial at 0 up to order/degree $k$) for $k$ sufficiently large. I don't understand this definition. They claim that that this is well-defined by Tougeron's theorem (which I would also be glad to know precisely which one of Tougeron's theorem, searching gives me several such theorems).
I think I would mostly appreciate and understand it if I have some "visual" description of modality. Or is this term something that is more of an algebraic/analytic nature?
