From the Thesis of B.Hassett I seem to understand that a smooth cubic 4-fold $X$ containing a $\mathbb{P}^2$ should contain also a octic K3, but I cannot see a natural way by which this K3 octic could appear inside $X$. How do you see that?


One wild guess is: via linkage of a quartic 3-fold that contains a quadric surface contained in $X$... but this doesn't seem very consistent...



octic K3s inside cubic 4-folds
The $\mathbb{P}^{2}$ contained in your cubic fourfold $X$ is cut out by linear forms (say) $L\_{1},L\_{2},$ and $L\_{3}.$ Since the homogeneous ideal of $X$ is contained in the homogeneous ideal generated by $L\_{1},L\_{2}$ and $L\_{3},$ there exist quadrics $Q\_{1},Q\_{2},$ and $Q\_{3}$ such that $X=\{L\_{1}Q\_{1}+L\_{2}Q\_{2}+L\_{3}Q\_{3}=0\}.$ The octic K3 cut out by $Q\_{1},Q\_{2}$ and $Q\_{3}$ is easily seen to lie in $X.$


