[@DC] that $r=r_{\lambda}$ is the normalization of $X_{\lambda}$. Notice that the projective coordinate ring of $X_{\lambda}\subset
{\mathbb P}({E({{\lambda}})})$ is given by the graded subalgebra $B({\lambda})=\bigoplus_{n=0}^\infty
 B_n({\lambda})$ of $A({\lambda})$ generated by ${E({{\lambda}})}^* \subset {\Gamma}(M,{\mathcal M}_{\lambda})$.

The variety $X_\Sigma$ {#ssez:XSigma}
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We consider now a generalization of the variety $X_{\lambda}$. Let $\Sigma$ be a finite set of dominant weights and denote ${E({\Sigma})} =
\bigoplus_{\mu \in \Sigma} {E({\mu})}$; let $x_\Sigma=[(I_\mu)_{\mu\in\Sigma}]\in {\mathbb P}({E({\Sigma})})$ and define $X_\Sigma$ as the closure of the $G\times G$-orbit of $x_\Sigma$ in ${\mathbb P}({E({\Sigma})})$. If $\Sigma = \{{\lambda}\}$, then we get the variety $X_{\lambda}$, while if ${\Sigma}= {\Pi^+({{\lambda}})}$ we get its normalization $\widetilde X _{\lambda}$. Notice that the diagonal action of $G$ fixes the point $x_\Sigma$ so we have a $G\times G$ equivariant map $G{\longrightarrow}X_\Sigma$ given by $g \longmapsto (g,1)x_\Sigma$. This map induces a map from $G_{\mathrm{ad}}$ to $X_\Sigma$ if and only if the action of the center of $G\times G$ on $E({\lambda})$ is the same for all ${\lambda}\in
\Sigma$ or equivalently if ${\Sigma}$ is contained in a coset of ${\Lambda}$ modulo ${\mathbb Z}{\Delta}$. In this case we say that $X_\Sigma$ is a *semi-compactification* of $G_{ad}$. If $G_{\mathrm{ad}}$ is a simple group and and $\Sigma \neq \{ 0 \}$ then 