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The configuration space of $n$ marked points on the complex plane isconsidered. We investigate a decomposition of this space by so-calledGauss-skizze i.e. a class of graphs being forests, introduced by Gauss. It isproved that this decomposition is a semi-algebraic topological stratification.It also forms a cell decomposition of the configuration space of $n$ markedpoints. Moreover, we prove that classical tools from deformation theory, ruledby a Maurer--Cartan equation, can be used only locally for Gauss-skizze. Weprove that the deformation of the Gauss-skizze is governed by aHamilton--Jacobi differential equation. This gives developments concerningSaito's Frobenius manifold. Finally, a Gauss-skizze operad is introduced. It isan enriched Fulton--MacPherson operad, topologically equivalent to the little2-disc operad.

Cech cover of the complement of the discriminant variety. Part II: Deformations of Gauss-skizze