On the singularities of Mishchenko-Fomenko systems

To each complex semisimple Lie algebra $\mathfrak{g}$ and regular element $a\in {\mathfrak{g}}_{\text{reg}}$, one associates a Mishchenko-Fomenko subalgebra ${\mathcal{F}}_a\subseteq \mathbb{C}[\mathfrak{g}]$. This subalgebra amounts to a completely integrable system on the Poisson variety $\mathfrak{g}$, and as such has a bifurcation diagram ${\mathrm{\Sigma }}_a\subseteq \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}({\mathcal{F}}_a)$. We prove that ${\mathrm{\Sigma }}_a$ has codimension one in $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}({\mathcal{F}}_a)$ if $a\in {\mathfrak{g}}_{\text{reg}}$ is not nilpotent, and that it has codimension one or two if $a\in {\mathfrak{g}}_{\text{reg}}$ is nilpotent. In the nilpotent case, we show each of the possible codimensions to be achievable. Our results significantly sharpen existing estimates of the codimension of ${\mathrm{\Sigma }}_a$.