Modality of a singularity

Remarks

Informally, a family of orbits depending on $m$ parameters is the union of the orbits of the points of an $m$-dimensional smooth submanifold of $J^k$. The modality does not depend on the choice of $k$, as long as $f$ is $k$-determined. Example: $f = x^3$ on $\C$ is simple. Any small perturbation of the jet of $x^3$ in $J^3$ is $a_1 x + a_2 x^2 + a_3 x^3$ with $a_3 \ne 0$. If $a_1 \ne 0$ it is right equivalent to $x$, a regular point. If $a_1 = 0 \ne a_2$ it is right equivalent to $x^2$, a Morse point. If $a_1 = a_2 = 0$ it is right equivalent to $x^3$. Non-example: $f = x^4 + y^4$ on $\C^2$ is not simple. For the germs $x^4 + y^4 + a x^2 y^2$ with $a^2 \ne 4$, the zero set is four distinct lines through $0$. Their cross-ratio, taken up to the order of the lines, depends on $a$ and is invariant under right equivalence. So infinitely many orbits meet every neighbourhood of $f$, and in fact the modality of $f$ is $1$.