If $X$ is a smooth projective variety of dimension at least $3$ over $\mathbb{C}$, Lefschetz's Hyperplane theorem says that for every hyperplane section $H$
$$\pi^1(H)\to\pi^1(X)$$
is an isomorphism, where $\pi^1$ can be taken to be the topological fundamental group or the étale one (as it is the profinite completion of the topological).
The same holds for the étale fundamental group over any field in any characteristic (using Leff conditions, e.g. SGA 2, XII Cor. 3.5), when $X$ is a smooth projective variety and $H$ is a regular hyperplane section.
Over the complex numbers, though, more is true: if $X$ is any smooth quasi-projective variety (of dimension at least $3$) and $H$ is a general hyperplane section, then
$$\pi^1(H)\to\pi^1(X)$$
is again an isomorphism, as far as I know this is proven using Morse Theory (see e.g. Sec. 5.1 in Goresky and MacPherson "Stratified Morse Theory").
My question is:
Could one expect a similar statement in positive characteristic? Is there a counterexample?
Note that the tame part of the fundamental group should not pose any problem, as there is a Lefschetz theorem for it (once we have a good compactification).
EDIT: I always thought that generic is commonly used for "at the generic point", general for "every rational point of something containing an open" and very general for "every rational point of something dense" (in this case, in the projective space parametrizing hyperplanes). Hence by general I meant for every hyperplane in some open of the parametrizing space.
Best Answer
This won't work for arbitrary smooth quasiprojective varieties. Take $X = \mathbb A^n$, then certainly any $H = \mathbb A^{n-1}$,.
But $\pi_1(\mathbb A^{n-1}) \to \pi_1(\mathbb A^n)$ is not an isomorphism in characteristic $p$ because there are nontrivial finite etale coverings of $\mathbb A^1$, which when pulled back to $\mathbb A^n$ become nontrivial coverings of $\mathbb A^n$ that trivialize when restricted to $\mathbb A^{n-1}$.