, If X has a Zariski-open covering by horospherical G-stacks, then X is a horospherical Gstack
That is, we may and do assume that T = P/H is trivial, i.e., H = P . Moreover, replacing G by a finite étale cover if necessary, we may and do assume that G is a direct product of a torus and a simply-connected semisimple group ,
, It follows from Corollary 5.3 that Z is a G-stable closed substack of codimension at least 2 in X . Let Y be the complement of Z in X . Then, Y is a G-stable dense open substack of X such that codim(G.y, Y) is at most 1 for all y ? Y(k), Define Z as the union of all closed substacks G.x, where x in X (k) runs over all points such that G.x is of codimension at least 2
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