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Geometrical description of smooth projective symmetric varieties with Picard number one. (English) Zbl 1194.14071
Let \(G/H\) be a symmetric space and let \(X\) be a symmetric variety, completion of \(G/H\). In a previous preprint [Smooth projective symmetric varieties with Picard number equal to one, arXiv:math/0702340, to appear in Int. J. Math.], the author classified all the symmetric varieties whose Picard number is one. In particular, it turns out that these varieties \(X\) are projective and Fano. In the paper under review, the author provides new information on the geometric structure of such varieties. In particular, using again the theory of Luna-Vurst colored fans, the author determines the automorphism group of \(X\). When \(X\) is not homogeneous, the author determines \(G\)-equivariant embeddings of \(X\) inside a homogeneous variety.

MSC:
14M17 Homogeneous spaces and generalizations
14L30 Group actions on varieties or schemes (quotients)
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