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 $\Sigma_a\subseteq\mathrm{Spec}(\mathcal{F}_a)$. We prove that $\Sigma_a$ has codimension one in $\mathrm{Spec}(\mathcal{F}_a)$ 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 $\Sigma_a$.