The genitive most commonly limits the meaning of substantives, adjectives, and adverbs, less commonly that of verbs.
Since the genitive has absorbed the ablative it includes (1) the genitive proper, denoting the class to which a person or thing belongs, and (2) the ablatival genitive.
a. The name genitive is derived from casus genitivus, the case of origin, the inadequate Latin translation of γενικὴ πτῶσις
Herbert Weir Smyth [n.d.], A Greek Grammar for Colleges; Machine readable text [info] [word count] [Smyth].