The non-self-intersection criterion for rigid modules over 2-complete quivers

Let $R$ be the path algebra of a 2-complete quiver, and let $f$ be the bijection from indecomposable modules corresponding to positive real roots to admissible curves on the associated Riemann surface. For an indecomposable $R$-module $M$, the non-self-intersection criterion.

$$\operatorname{Ext}^1(M,M)=0 \quad\Longleftrightarrow\quad f(M)\text{ has no self-intersections}.$$

Here, $\operatorname{Ext}^1(M,M)=0$ means that $M$ is rigid, while the curve model translates rigidity into the absence of self-intersections. The paper later proves this criterion for 2-complete rank-$3$ quivers; the general 2-complete case is attributed in the source to subsequent work of Felikson and Tumarkin.