Abstract
We study equivariant embeddings with small boundary of a given homogeneous space G / H , where G is a connected linear algebraic group with trivial Picard group and only trivial characters, and H ⊂ G is an extension of a connected Grosshans subgroup by a torus. Under certain maximality conditions, like completeness, we obtain finiteness of the number of isomorphism classes of such embeddings, and we provide a combinatorial description of the embeddings and their morphisms. The latter allows a systematic treatment of examples and basic statements on the geometry of the equivariant embeddings of a given homogeneous space G / H .
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