Lefschetz on étale fundamental group for quasi-projective varieties

Question

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.

Answer 1

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 $\mathbb A^{n-1}$.

Answer 2

I am posting my comments above as an answer, together with a reference (per the OP's request). The difference between Will's answer and my answer results from the ambiguous term "general". Only the OP can specify what is meant by "general" in the question. In this answer, I interpret "general" as meaning "the property is true for the geometric generic point".

Let $k$ be an algebraically closed field. Let $X$ be an integral $k$-scheme of dimension $\geq 2$, and let $f:X\to \mathbb{P}^n_k$ be a finite type morphism that is unramified on a dense, open subscheme of $X$. Denote by $(\mathbb{P}^n_k)^\vee$ the projective space parameterizing hyperplanes in $\mathbb{P}^n_k$. In particular, for every algebraically closed field extension $\kappa/k$, there is a natural bijection between the hyperplanes $H\subset \mathbb{P}^n_\kappa$ and the $k$-morphisms $[H]:\text{Spec}(\kappa) \to (\mathbb{P}^n_k)^\vee$.

Let me restate Corollary 2.2 of the following (which, in turn, depends on Théorème 4.10 and 6.10 of Jouanolou's "Théorèmes de Bertini et applications").

MR3114946
Graber, Tom; Starr, Jason Michael
Restriction of sections for families of abelian varieties.
A celebration of algebraic geometry, 311–327,
Clay Math. Proc., 18, Amer. Math. Soc., Providence, RI, 2013.

For every generically finite morphism $g:Y\to X$ that admits no rational section, there exists a dense, Zariski open subscheme $U\subset (\mathbb{P}^n_k)^\vee$ such that for every algebraically closed field $\kappa$ and for every geometric point $[H]:\text{Spec}(\kappa) \to U$, also $Y\times_{\mathbb{P}^n_k} H \to X\times_{\mathbb{P}^n_k} H$ admits no rational section.

Now let $K$ denote the algebraic closure of the function field of $(\mathbb{P}^n_k)^\vee$ as an extension of $k$, and denote by $[H]:\text{Spec}(K)\to (\mathbb{P}^n_k)^\vee$ the corresponding $k$-morphism. Then for every connected, finite, etale morphism $g:Y\to X$, also the morphism $Y\times_{\mathbb{P}^n_k} H \to X\times_{\mathbb{P}^n_k} H$ is connected, finite and etale. Therefore, the induced homomorphism, $$\pi^1(X\times_{\mathbb{P}^n_k} H) \to \pi^1(X),$$ is surjective.