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Simple immersions of wonderful varieties. (English) Zbl 1122.14036

Given a complex connected semisimple linear algebraic group \(G\), the author gives a classification of all wonderful \(G\)-varieties which can be realized as an equivariant closed subvariety of the projective space of a simple \(G\)-module. In fact, it is shown that if \(X\) is a wonderful \(G\)-variety, then such a closed immersion exists if and only if all the isotropy groups in \(X\) equal their normalizers, and if this is the case, then the immersion is unique.
A wonderful \(G\)-variety is a smooth complete \(G\)-variety having an open (dense) orbit \(O\) such that the complement of \(O\) is a union of smooth prime divisors \(D_i\), \(i\in\{1,\ldots,n\}\) with normal crossings, such that the intersection \(\bigcap_{i=1}^n D_i\) is non empty, and such that two points of \(X\) are in the same orbit if and only if they are in the same set of divisors \(D_i\). It was shown by D. Luna [Transform. Groups 1, No. 3, 249–258 (1996; Zbl 0912.14017)] that all wonderful varieties are spherical, and thus the theory of spherical varieties can be used to study this problem. The method used is to reduce the problem to the finite set of rank one wonderful varieties which are not obtained by parabolic induction. Here, the result is checked directly on each case. Finally, the same techiques are used to show that all ample divisors of wonderful varieties are very ample.

MSC:

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

Citations:

Zbl 0912.14017
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References:

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