On the Commuting variety of a reductive Lie algebra and other related varieties.
The nilpotent cone of a reductive Lie algebra has a desingularization given by the 
cotangent bundle of the flag variety. Analogously, the nullcone of a cartesian 
power of the algebra has a desingularization given by a vector bundle over the 
flag variety. As for the nullcone, the subvariety of elements whose components 
are in a same Borel subalgebra, has a desingularization given by a vector bundle over 
the flag variety. In this note, some properties of these varieties are given. For 
the study of the commuting variety, the analogous variety to the flag variety is 
the closure in the Grassmannian of the set of Cartan subalgebras. So some 
properties of this variety are given. In particular, it is smooth in codimension $1$. 
We introduce the generalized isospectral commuting varieties and give some properties. 
Furthermore, desingularizations of these varieties are given by fiber bundles over a 
desingularization of the closure in the grassmannian of the set of Cartan subalgebras 
contained in a given Borel subalgebra.