Variation of Mumford quotients by torus actions on full flag varieties. I

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 $A_n$.