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On complete orbit spaces of SL(2) actions. II. (English) Zbl 0813.14032

[For part I see ibid. 55, No. 2, 229-243 (1988; Zbl 0682.14034).]
The aim of this paper is to extend the results of part I (loc. cit.) concerning geometric quotients of actions of SL(2) to the case of good quotients.
Conjecture. Let \(X\) be a smooth algebraic variety with an action of a reductive group \(G\). Let \(T\) be a maximal torus of \(G\) and let \(N(T)\) be its normalizer in \(G\). Let \(U\) be an \(N(T)\)-invariant open subset of \(X\) for which there exists a good quotient \(\pi : U \to U // T\) and which is maximal with respect to this property. Then \(\bigcap_{g \in G} gU\) is open, \(G\)-invariant and there exists a good quotient \(\bigcap_{g \in G} gU \to \bigcap_{g \in G} gU // G\). Moreover, if \(U// T\) is complete, then \(\bigcap_{g \in G} gU // G\) is also complete.
In the present paper we only consider the case \(G = SL(2)\). Theorem 1 shows that if \(U // T\) is projective then the conjecture is valid. Moreover, then \(X\) and \(\bigcap_{g \in SL (2)} gU // G\) are projective and there exists an ample, invertible, \(G \)-linearized sheaf \({\mathcal L}\) on \(X\) such that \(U\) is the set of semi-stable points with respect to the action of \(T\) induced by the action of \(G\). – We also prove the conjecture under the additional assumption that either \(U // T\) is complete (theorem 2) or \(U // T\) is quasi-projective (theorem 9). – Answering a question of D. Luna we also describe an example of an action of SL(2) on an algebraic variety \(X\) such that there exists a geometric quotient \(X \to X/SL(2)\), where \(X/SL (2)\) is an algebraic space but not an algebraic variety.

MSC:

14L30 Group actions on varieties or schemes (quotients)
14M17 Homogeneous spaces and generalizations
20G15 Linear algebraic groups over arbitrary fields

Citations:

Zbl 0682.14034
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