  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 $\ge 2$.
