adjoint group (see section \[ssez:XSigma\]). We will make use of the wonderful compactification of $G_\mathrm{ad}$, the adjoint group of $G$, and of the results on projective normality of these compactifications proved by S. Kannan in [@Ka]. These results hold in the more general case of a symmetric variety; however our method does not apply to this more general situation (see section \[ssez:simmetrichenormalita\]).

\[see Theorem \[smooth Xsigma\]\] The variety $X_{\lambda}$ is smooth if and only if $\lambda$ satisfies property $(\star)$ of Theorem A together with the following properties:

-   For every connected component $\Delta'$ of $\Delta$, $\operatorname{Supp}(\lambda)\cap \Delta'$ is connected and, in case it contains a unique element, then this element is an extreme of ${\Delta}'$;

-   $\operatorname{Supp}(\lambda)$ contains every simple root which is adjacent to three other simple roots and at least two of the latter;

-   Every connected component of ${\Delta}{\smallsetminus}\operatorname{Supp}(\lambda)$ is of type ${\mathsf A}$.

Theorem B can be generalized to any simple and normal adjoint symmetric variety. Following a criterion of ${\mathbb Q}$-factoriality for spherical varieties given by M. Brion in [@Br2], properties i) and ii) characterize the $\mathbb{Q}$-factoriality of the normalization of $X_{\lambda}$ (see Proposition \[Q-fattorialita\]), while property iii) arises from a criterion of smoothness given 