Poisson blow-ups and the adjoint quotient

We leverage Polishchuk's Poisson blow-up criterion in the context of algebro-geometric integrable systems. In more detail, one may associate an integrable system $τ:\mathfrak{X}\longrightarrow\mathfrak{B}$ to each affine Poisson scheme $\mathfrak{X}$ over $\mathbb{C}$. We prove that the blow-ups of $\mathfrak{X}$ along fibers of $τ$ are Poisson schemes occurring in a family $\widetilde{\mathfrak{X}\times\mathfrak{B}}\longrightarrow\mathfrak{B}$, where $\widetilde{\mathfrak{X}\times\mathfrak{B}}$ is itself a Poisson scheme. This result is subsequently specialized to the adjoint quotient $τ:\mathfrak{g}\longrightarrow\mathfrak{g}/\!/G=:\mathfrak{c}$ of a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$ with integrating algebraic group $G$. We show that the family $\widetilde{\mathfrak{g}\times\mathfrak{c}}\longrightarrow\mathfrak{c}$ is flat, conical, and equipped with a canonical Poisson Hamiltonian $G$-variety structure. We also obtain Poisson-geometric results on the fibers of this family, which are blow-ups of $\mathfrak{g}$ along regular adjoint orbit closures.