If $Z\subset \mathbb{P}_{\mathbb{C}}^{n+1}$ is a variety of dimension greater than $\frac{n}{2}$ and $X\subset\mathbb{P}_{\mathbb{C}}^{n+1}$ is a smooth hypersurface of degree $d$ containing $Z$, then an application of Lefschetz hyperplane theorem shows $\deg(X)|\deg(Z)$. This is given in Corollary C.10 of 3264 & All That by Harris and Eisenbud.
In addition, the same corollary claims that in particular, if $Z$ is of prime degree $p$ and nondegenerate, then it is not strictly contained in a smooth hypersurface.
How do we rule out the case where $\deg(X)=\deg(Z)=p$?
Edit: cross posted in MO in a different form (https://mathoverflow.net/questions/251504/hypersurface-containing-nondegenerate-subvariety-of-same-degree-and-large-dimens)