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 $\g$ 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 $\g$ [Suslin-Friedlander-Bendel, J. Amer. Math. Soc, \textbf{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 $C_r(\mathfrak{gl}_2), C_r(\fraksl_2)$ and $C_r(\N)$ where $\N$ is the nilpotent cone of $\fraksl_2$. Our calculations lead us to state a conjecture on Cohen-Macaulayness for commuting varieties of $r$-tuples. Furthermore, in the case when $\g=\fraksl_2$, we obtain interesting results about commuting varieties when adding more restrictions into each tuple. In the case of $\fraksl_3$, we are able to verify the aforementioned properties for $C_r(\fraku)$. Finally, applying our calculations on the commuting variety $C_r(\overline{\calO_{\sub}})$ where $\overline{\calO_{\sub}}$ is the closure of the subregular orbit in $\fraksl_3$, we prove that the nilpotent commuting variety $C_r(\N)$ has singularities of codimension $\ge 2$.