Suppose $X\subset \mathbb{P}^n$ is a smooth hypersurface defined over $\mathbb{Q}$. For a "generic" prime $p$, what can be said about the set of hyperplanes $H$ in $\mathbb{P}^n(\mathbb{F}_p)$ for which $H \cap X$ is smooth over $\mathbb{F}_p$? For $p$ fixed and $X$ varying, by contrast, the situation can be arbitrarily bad: in fact, every hyperplane section of

${\sum_{i=1}^{n+1}X_i X_{i+n+1}^p=0} \subset \mathbb{P}^{2n+1}$ over $\mathbb{F}_p$ is singular.