Let \(G\) be a semisimple algebraic group over an algebraically closed field of characteristic 0. A \(G\)-variety is spherical if it is normal and if it contains an open \(B\)-orbit, where \(B\subset G\) is a Borel subgroup; in this case there are a finite number of \(B\)-orbits (resp. of \(G\)-orbits). Examples of spherical varieties are the symmetric varieties and the toric varieties (if one allow \(G\) to be only reductive). Homogeneous symmetric varieties are spaces \(G/H\), where \(G^{\sigma}\subset H\subset N(G^{\sigma})\) and \(G^{\sigma}\) is the set of fixed points by an involution \(\sigma\) of \(G\). Another important class of spherical varieties are the wonderful varieties, namely smooth projective \(G\)-varieties \(M\) with an open orbit and such that: i) the complement of the open orbit is a normal crossing divisor with smooth components; ii) the closure of the orbits are the intersections of such components; iii) the intersection of all these components is the unique closed orbit of \(M\). In particular, flag varieties are wonderful. Moreover, any spherical homogeneous space \(G/H\) with \(H\) self-normalizing has a (unique) wonderful completion by C. De Concini and C. Procesi [in: Invariant theory, Proc. 1st 1982 Sess. C.I.M.E., Montecatini/Italy, Lect. Notes Math. 996, 1–44 (1983; Zbl 0581.14041)] in the symmetric case and by F. Knop, [J. Am. Math. Soc. 9, No. 1, 153–174 (1996; Zbl 0862.14034)] in the general case.
However, there are many natural examples of completions of a spherical homogeneous space which do not need to be normal. For example, consider the projective space \( \mathbb{P}(V)\) associated to a simple \(G\)-module \(V\). P. Bravi and D. Luna [J. Algebra 329, No. 1, 4–51 (2011; Zbl 1231.14040)] proved that any \(G\)-orbit \(G[v]\) in \(\mathbb{P}(V)\) has a wonderful completion \( M\) and that such completion dominates the closure \(X\) of \(G[v]\) in \(\mathbb{P}(V)\). Thus, \(M\) dominates also the normalization \(\widetilde{X}\) of \(X\). In general \(X\) is non-normal; however, in [D. A. Timashev, Sb. Math. 194, No. 4, 589–616 (2003); translation from Mat. Sb. 194, No. 4, 119–146 (2003; Zbl 1074.14043)] it has been proved that there is a one-to-one correspondence between the orbits of \(X\) and the orbits of \(\widetilde{X}\). Moreover, A. Maffei has proved [Transform. Groups 14, No. 1, 183–194 (2009; Zbl 1185.14044)] that, when \(G/H\) is symmetric, the normalization morphism \(q: \widetilde{X}\rightarrow X\) is a bijective. The author of this work is interested to understand the bijectivity of \(q\) when \(G/H\) is only spherical.
First, the author gives a description of the set of orbits of \(X\) (and \(\widetilde{X}\)) by using the morphism \(M \rightarrow\widetilde{X}\rightarrow X\). This leads to a combinatorial criterion to establish whether or not two orbits in \(M\) map onto the same orbit on \(X\) which, in particular, implies that different orbits in \(X\) are never \(G\)-equivariantly isomorphic. Moreover, he gives a combinatorial criterion so that the restriction of \(q\) to a fixed orbit of \(\widetilde{X}\) is an isomorphism. This criterion can be simplified when \(M\) is strict, i.e. all the isotropy subgroup of \(M\) are self-normalizing. The strict wonderful varieties, introduced by G. Pezzini [Math. Z. 255, No. 4, 793–812 (2007; Zbl 1122.14036)], are those wonderful varieties which can be embedded in a simple projective space. This leads the author to prove the main result of this work: a combinatorial criterion for \(q\) to be bijective under the assumption that \(M\) is strict. It is given in term of the spherical system of \(M\) (and of the \(B\)-stable divisor which gives the morphism \(M\rightarrow X\subset\mathbb{P}(V)\)). The condition of bijectivity involves the double links of the Dynkin diagram of \(G\) and it is trivially fulfilled whenever \(G\) is simply laced or \(M\) is symmetric. The main examples of strict wonderful varieties where bijectivity fails arise from the context of wonderful model varieties introduced by D. Luna [J. Algebra 313, No. 1, 292–319 (2007; Zbl 1116.22006)]. A model space for \(G\) is a quasi-affine homogeneous space whose coordinate ring contains every simple \(G\)-module with multiplicity 1. In the previous work are classified the model spaces and it is introduced the wonderful model variety, whose orbits naturally parametrize, up to isomorphism, the model spaces for \(G\): every orbit of this last variety is of the shape \(G/N(H)\) where \(G/H\) is a model space.