Geodesic
Variation of Mumford quotients by torus actions on full flag varieties. I
Izvestiya. Mathematics , Tome 71 (2007) no. 6, pp. 1105-1122

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We study the variation of the Mumford quotient by the action of a maximal torus $T$ on a flag variety $G/B$ as we change the projective embedding $G/B \hookrightarrow\mathbb P(V(\chi))$, where the $T$-linearization is induced by the standard $G$-linearization. To do this, we describe the linear spans of the supports of the semistable orbits. This enables us to calculate the rank of the Picard group of the quotient $(G/B)^{ss}//T$ in the case when $G$ contains no simple components of type $A_n$. 
@article{IM2_2007_71_6_a1,
     author = {V. S. Zhgoon},
     title = {Variation of {Mumford} quotients by torus actions on full flag varieties. {I}},
     journal = {Izvestiya. Mathematics },
     pages = {1105--1122},
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     volume = {71},
     number = {6},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2007_71_6_a1/}
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V. S. Zhgoon. Variation of Mumford quotients by torus actions on full flag varieties. I. Izvestiya. Mathematics , Tome 71 (2007) no. 6, pp. 1105-1122. http://geodesic.mathdoc.fr/item/IM2_2007_71_6_a1/