Commuting varieties of $r$-tuples over Lie algebras

Let $G$ be a simple algebraic group defined over an algebraically closed field $k$ of characteristic $p$ and let $\text{g}$ be the Lie algebra of $G$. It is well known that for $p$ large enough the spectrum of the cohomology ring for the $r$-th Frobenius kernel of $G$ is homeomorphic to the commuting variety of $r$-tuples of elements in the nilpotent cone of $\text{g}$ [Suslin-Friedlander-Bendel, J. Amer. Math. Soc, $\mathbf{\text{10}}$ (1997), 693–728]. In this paper, we study both geometric and algebraic properties including irreducibility, singularity, normality and Cohen-Macaulayness of the commuting varieties $C_r({\mathfrak{g}\mathfrak{l}}_2),C_r({\text{fraksl}}_2)$ and $C_r(\mathbb{N})$ where $\mathbb{N}$ is the nilpotent cone of ${\text{fraksl}}_2$. Our calculations lead us to state a conjecture on Cohen-Macaulayness for commuting varieties of $r$-tuples. Furthermore, in the case when $\text{g}={\text{fraksl}}_2$, we obtain interesting results about commuting varieties when adding more restrictions into each tuple. In the case of ${\text{fraksl}}_3$, we are able to verify the aforementioned properties for $C_r(\text{fraku})$. Finally, applying our calculations on the commuting variety $C_r(\overline{{\text{calO}}_{\subset }})$ where $\overline{{\text{calO}}_{\subset }}$ is the closure of the subregular orbit in ${\text{fraksl}}_3$, we prove that the nilpotent commuting variety $C_r(\mathbb{N})$ has singularities of codimension $\ge 2$.