  Fix a rank g free group F and a connected reductive complex algebraic group
G. Let X(F,G) be the G-character variety of F. When the derived subgroup DG in
G is simply connected we show that X(F,G) is factorial (which implies it is
Gorenstein), and provide examples to show that when DG is not simply connected
X(F,G) need not even be locally factorial. Despite the general failure of
factoriality of these moduli spaces, using different methods, we show that
X(F,G) is always Gorenstein.
