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The purpose of this paper is to study the local structure of thesemi-invariant picture of a tame hereditary algebra near the null root. Using aconstruction that we call co-amalgamation, we show that this local structure iscompletely described by the semi-invariant pictures of a collection ofself-injective Nakayama algebras. We then describe the cones of this localstructure using cluster-like structures that we call support regular clusters.Finally, we show that the local structure is (piecewise linearly) invariantunder cluster tilting.

Infinitesimal semi-invariant pictures and co-amalgamation