ABSTRACT
This chapter introduces the Configuration varieties defined by typical graphs. The galleries of types, or more generally the linear typical graphs, define a class of k-smooth and integral Configurations varieties particularly important in this work. A minimal generalized gallery of the Flag Complex is a general point of the Configurations variety given by its gallery of types. The configurations variety associated to the graph is a smooth resolution of singularities such that the pull-back of the tangent submodule admits an extension as a locally trivial submodule. The generalized gallery may be completed into an adapted gallery so that: as it follows from some equalities. Thus one has necessarily that length, and that is a minimal gallery. It is concluded that for all maximal length flag , at maximal distance from d, the generalized gallery is minimal.
