Effective cycles on universal hypersurfaces

We study the effective cones of cycles on universal hypersurfaces on a projective variety $X$, particularly focusing on the case of universal hypersurfaces in ${\mathbb{P}}^n$. We determine the effective cones of cycles on the universal conic over ${\mathbb{P}}^2$. We also determine every effective cone of cycles of dimension at most 6 or equal to 10 on any universal plane curve.