By D.A. Timashev

Homogeneous areas of linear algebraic teams lie on the crossroads of algebraic geometry, thought of algebraic teams, classical projective and enumerative geometry, harmonic research, and illustration concept. by way of average purposes of algebraic geometry, with the intention to clear up a variety of difficulties on a homogeneous house, it really is normal and beneficial to compactify it whereas keeping an eye on the gang motion, i.e., to contemplate equivariant completions or, extra often, open embeddings of a given homogeneous house. Such equivariant embeddings are the topic of this ebook. We specialise in the class of equivariant embeddings by way of convinced information of "combinatorial" nature (the Luna-Vust thought) and outline of assorted geometric and representation-theoretic homes of those kinds in line with those facts. the category of round types, intensively studied over the past 3 a long time, is of precise curiosity within the scope of this publication. round forms contain many classical examples, similar to Grassmannians, flag types, and forms of quadrics, in addition to famous toric kinds. we've got tried to hide lots of the very important concerns, together with the new vast development acquired in and round the idea of round varieties.