A finiteness condition for algebras over coordinate rings of varieties

Let A be a finitely presented k[X]-algebra, where k[X] is the algebra of regular functions on a variety X defined over an algebraically closed field k. The following problem arises in the study of degenerations of algebras (Hajduk and Kasjan, in press [7]). Assume that the specialization of A at x has finite dimension over k for every x from a dense subset of X. Is there an open dense subset U of X such that the localization of A with respect to U is a finitely generated k[U]-module? We prove that this is the case if k has infinite transcendence degree over its prime subfield. We provide some applications to the concept of generalized CB-degenerations.