The GIT-equivalence for \(G\)-line bundles. (English) Zbl 0955.14035

Ressayre, N.

Geom. Dedicata 81, No. 1-3, 295-324 (2000).

The author sharpens and clarifies some of the results in the theory of variation of GIT-quotients for a reductive linear group \(G\) as developed in the work by M. Thaddeus [J. Am. Math. Soc. 9, No. 3, 691-723 (1996; Zbl 0874.14042)] and I. Dolgachev and Y. Hu [Publ. Math., Inst. Hautes Étud. Sci. 87, 5-56 (1998)]. In particular he shows that the GIT-classes (the sets of algebraic equivalence classes of linearized line bundles \(L\) defining the same set of semi-stable points) are equal to the relative interiors of rational polyhedral convex cones \(C_i\) which form a finite fan in the \(G\)-ample cone.
The terminology in the paper under review follows the terminology of the paper by Dolgachev and Hu (loc. cit.), however there are slight modifications. For example, a wall is assumed to be always of codimension one, and a chamber is a connected component of the complement of the union of the walls. An example from the appendix to the Dolgachev-Hu paper written by the author shows that a (new) chamber could be an (old) wall of codimension 0. The convex cones \(C_i\) are the relative interiors of the faces of chambers. The theory is illustrated in the special case when \(G = k^*\times SL(2)\).

Reviewer: Igor V.Dolgachev (Ann Arbor)

MSC:

14L24	Geometric invariant theory
14M17	Homogeneous spaces and generalizations
14L30	Group actions on varieties or schemes (quotients)
14E30	Minimal model program (Mori theory, extremal rays)

Keywords:

quotients for a reductive linear group; GIT-classes; wall; chambers; geometric invariant theory