# zbMATH — the first resource for mathematics

Plongements d’espaces homogènes. (English) Zbl 0545.14010
Given a homogeneous space $$G/H$$ (G a connected algebraic group over an algebraically closed field k of characteristic 0, H a closed subgroup), an embedding is an irreducible G-variety X together with an open G equivariant immersion $$G/H\hookrightarrow X$$. In this paper the authors start a general theory of embeddings. Their point of view is rational. Thus they begin by reformulating the notion of embedding in these terms (definition 1.5). They then pass to make a detailed study of what they define as an elementary embedding, that is a normal embedding having only two orbits, the open orbit and an orbit of codimension one. In the point of view taken the set of elementary embeddings is in one-one correspondence with the set $${\mathcal V}_ 1(G/H)$$ of G invariant normalized geometric discrete valuations v with $$k^ G_ v=k$$. Using this the authors associate to a formal curve $$\lambda \in((G/H)_{k((t))}-(G/H)_{k[[t]]})=(G/H)^*_{k((t))}$$ an elementary embedding $$X_{\lambda}$$, and show that each elementary embedding is an $$X_{\lambda}$$ for a suitable $$\lambda \in(G/H)^*_{k((t))}$$ (actually their result is much more precise). - $$X_{\lambda}$$ has the property that given any embedding X, an orbit T in X such that $$\lim_{t\to 0}\lambda(t)$$ exists and lies in T one has a dominant G equivariant morphism $$X_{\lambda}\to X$$ mapping the codimension one orbit in X onto T. Furthermore one has that each G invariant geometric discrete valuation of k(G/H) is a limit in a suitable sense of a sequence of elements in $${\mathcal V}_ 1(G/H)$$. The authors say that this fact can be effectively used in some cases to deduce the knowledge of the G invariant valuations from the knowledge of $${\mathcal V}_ 1(G/H)$$. In the last sections the authors assume that G is reductive and that the algebra of regular functions on G is factorial. They fix a Borel subgroup B in G and denote by $${\mathcal P}$$ the set of eigenvectors for B acting on k(G). They show that a G invariant discrete valuation of k(G/H) is determined by its value on $${\mathcal P}\cap k(G/H)$$. This is used to show that when B has a dense orbit in $$G/H$$ any embedding has finitely many orbits and in the general case to give a method to determine $${\mathcal V}_ 1(G/H)$$ which is ”usable” at least when $$tr \deg(k(G/H)^ B)\leq 1.$$- In section 8 the extra hypothesis that the embedding X is normal is made. Under this assumption the authors show that given an orbit $$T\subset X$$, the local ring $${\mathcal O}_{X,T}={\mathcal O}_{\ell}$$ is determined by the finite set of valuations in $${\mathcal V}(G/H)$$ essential for $${\mathcal O}_{\ell}$$ plus the set $$^ B{\mathcal D}_{\ell}$$ of essential valuations for $${\mathcal O}_{\ell}$$ associated to B stable divisors in G/H. Furthermore for a finite set $${\mathcal W}\subset {\mathcal V}(G/H)$$ and a set $${\mathcal D}$$ of discrete valuations associated to B stable divisors in G/H, necessary and sufficient conditions of a combinatorial flavour are given on $$({\mathcal W},{\mathcal D})$$ to arise in the above way. In the final section as an application of their theoy the authors classify the normal embeddings of Sl(2).
Reviewer: C.de Concini

##### MSC:
 14E25 Embeddings in algebraic geometry 14M17 Homogeneous spaces and generalizations
Full Text: