Let $X \subset \mathbb{P}^{2m+1}$ be a smooth hypersurface of degree $d$ and even dimension $2m$ containing a linear subspace $L \cong \mathbb{P}^m$, and let $h$ be the hyperplane class. Two intersection numbers are standard: $L \cdot h^m = 1$ and $h^m \cdot h^m = d$. If the class of $L$ were a rational multiple $c\,h^m$, then $c\,d = 1$ and $L \cdot L = c^2 d$ would be an integer, since intersection numbers of algebraic cycles are integers.

This concept isolates the arithmetic step: for every integer $d \ge 2$ there is no rational $c$ with $c\,d = 1$ and $c^2 d$ an integer. Hence $[L]$ is not a rational multiple of $h^m$ for any $d \ge 2$, in every even dimension.