Some properties of generalized cluster algebras of geometric types

We study the lower bound algebras generated by the generalized projective cluster variables of acyclic generalized cluster algebras of geometric types. We prove that this lower bound algebra coincides with the corresponding generalized cluster algebra under the coprimality condition. As a corollary, we obtain the dual PBW bases of these generalized cluster algebras. Moreover, we show that if the standard monomials of a generalized cluster algebra of geometric type are linearly independent, then the directed graph associated to the initial generalized seed of this generalized cluster algebra does not have 3-cycles.