Given a smooth, even dimensional algebraic variety X and a very ample line bundle L, associated to a primitive Hodge class z, there are several rational maps to Hodge-theoretically defined target spaces. These maps have the property that if the induced map on a particular cohomology class is non-zero then there exists an algebraic cycle Z with fundamental class z. A basic question is: If conversely a cycle Z exists, then is one of the induced cohomology maps non-zero if L is sufficiently ample relative to z? Even in the classical case when X is an algebraic surface, this question leads to new and interesting issues concerning the geometry of curves on X. It seems in this case that the question has an affirmative answer, provided that one lets the pair (X, z) vary in moduli. This talk will report on joint work in progress with Mark Green, picking up on a theme first discussed with Ron Donagi some 25 years ago.