“…To this end, we are going to use the explicit formulas, in terms of cross-ratios, given by Volkov in [Vol1] for the Y-variables of the Y-system Y(A m , A n ). As in the A n -case, there are natural birational identifications between Conf m+n+2 (P m ) and the X-cluster variety of type A m ⊠ A n (see [GGSVV,§6.3] and [Wen,§2.3]): given a graph Γ of a certain type (namely, a 'minimal bipartite graph' ), one constructs a birational map x Γ : Conf m+n+2 (P m ) X A m ⊠A n . For instance, one can (and we will) consider the graph Γ 0 of [Wen,Example 2.11].…”