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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

14E25 Embeddings in algebraic geometry
14M17 Homogeneous spaces and generalizations
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