  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.
