Counting using Hall Algebras II. Extensions from Quivers

We count the ${\mathbb{F}}_q$-rational points of GIT quotients of quiver representations with relations. We focus on two types of algebras – one is one-point extended from a quiver $Q$, and the other is the Dynkin $A_2$ tensored with $Q$. For both, we obtain explicit formulas. We study when they are polynomial-count. We follow the similar line as in the first paper but algebraic manipulations in Hall algebra will be replaced by corresponding geometric constructions.