The influence of Pt thickness on the interface structure (roughness / intermixing) and magnetic properties has been investigated for (Co / Pt) multilayers sputtered on a Pt or a thin oxide (MgO or AlO x ) buffer layer. When Pt thickness increases from 1.2 nm-2.2 nm, we observe that the effective anisotropy increases with the Pt thickness, simultaneously with the decrease of roughness, i.e. the occurrence of sharper interfaces. Perpendicular magnetic anisotropy (PMA) is still achieved on the oxide buffer layers, but with a lower effective anisotropy correlated to more perturbed interfaces. The detailed analysis of the saturation magnetization shows that: (i) M s is significantly enhanced in the case of rough/intermixed interfaces, which is attributed to and discussed in the framework of Pt induced polarization, (ii) the change in volume dipolar anisotropy is the main factor responsible for the reduction of K eff for systems grown on oxides. Beyond the major role of volume dipolar contribution that reduces PMA, a supplemental positive contribution promoting PMA can be invoked for rough interfaces and large M s (deposit on oxide). This contribution is consistent with a dipolar surface anisotropy term and increases for rough interfaces, in contrast to the Néel surface anisotropy. These opposite variations may interestingly lead to an enhanced anisotropy in (Co / Pt) stackings grown on oxides compared to systems deposited on Pt, i.e. with sharper interfaces.