I was reading the Phd Thesis of J. Montaldi and I came across the following paragraphs:
''In the cases where $k+d<\sigma(k, p)+p$ then we let $\left(W_1, \ldots, W_3\right)$ be the rinite set of $x$-orbits in $J^ r(X, \mathbb{R}^ P)$ of codimension less than $k+d$, and $\left(w_{s+1}, \ldots, w_t\right)$ a finite stratification of the complement of $w_1 \ldots W_s$. Then for the general situation described in theorem 2.2 and the particular extension in theorem 2.3 , let $\mathrm{R}$ be the intersection of the $R_W$, which will itself be a residual subset of $\operatorname{Imm}\left(X, R^n\right)$. For $g \in R$, the associated jet-extension map will then miss the $w_1$ for $i>s$, and be transverse to the $w_1$ for $16 \mathrm{~s}$, and such a mapping will be termed generic for the given situation. In our applications in chapters 4 and 5 , the hypothesis $k+d<\sigma(k, p)+p$ will indeed be fulfilled, and for those chapters, the term generic will have the meaning described in this paragraph.
If, on the other hand, $k+d \geqslant \sigma(k, p)+p$, then the meaning of the word generic should have is not so straightforward. However, in practice, it seems that one can always take the strata to be the union of the $\mathcal{K}$-orbits as the moduli (or modal parameters) vary, (usually excluding some exceptional values).
When the dimensions are such that the modal singularities are not encountered for a generic immersion, the transversality to the stratification ensures that all the singularities are presented transversely, (1.e. they are versally deformed). If, on the other hand, the modal singularities do occur, then the singularities cannot all be presented transversely. Thus in our applications, all the singularities that arise for a generic immersion will be presented transversely.''.
What bothers me is the last paragraph, when he says "that all the singularities cannot be presented transversely". Can someone clarify what is going on here? If I considered the stratification that is being suggested in paragraph 2, then all the singularities (simple and non-simple) should be presented transversely?