Is the map on tame fundamental groups of a quasi-projective variety, upon base change between algebraically closed fields, an isomorphism?

Question

$\DeclareMathOperator\Spec{Spec} $Let $k \subset L$ be two algebraically closed fields of characteristic $p$. Let $U \subset \mathbb P^1_k$ be a smooth quasi-projective curve and let $U_L$ denote the base change of $U$ to $\Spec (L)$. Does anyone have a reference for why the map on the tame fundamental groups $\pi_1^t(U_L) \rightarrow \pi_1^t(U)$ is an isomorphism?

What I'm interested in is that, supposedly the categories of tame finite étale covers over $U$ and $U_L$ should be equivalent. But I can't seem to find a reference in the literature.

Answer 1

update: there is now a complete reference [2005.09690] Invariance of the tame fundamental group under base change between algebraically closed fields.

I think it is generally admitted as folklore that your claim/guess is true, see the introduction of [1612.02154] Gieseker conjecture for homogeneous spaces by Giulia Battiston. But finding a reference at this level of generality is difficult. Working with a more general setup actually makes things easier. Let me sketch a possible approach:

Step 1 : replace your curve $U$ by the complement $U=X\backslash D$ of a normal, proper scheme $X$ over an algebraically closed field endowed with a normal crossings divisor $D$. In other words, work in any dimension, but fix a compactification (there is a canonical smooth compactification in dimension $1$).

Step 2 : see your formula as a special case of Künneth formula, that is, if $V$ is another compactified variety then $\pi_1^t(U\times_k V)\simeq \pi^t_1(U)\times \pi_1^t(V)$.

Step 3: replace your scheme $U$ by an object with is homotopically equivalent, but proper over $k$. Here (as a famous algebraic geometer once told me) logarithmic geometry begs to be employed, and you dispose of the log scheme $X_{log}$ associated to the pair $(X,D)$, whose fundamental group is precisely $\pi_1^t(U)$.

Step 4: show the Künneth formula for proper log schemes. This was done by Yuichiro Hoshi in The exactness of the log homotopy sequence Proposition 3, as mentionned in the introduction of Giulia Battiston's article above.

In step 3, if you are more inclined towards algebraic stacks, you could replace log schemes by infinite root stacks à la Talpo-Vistoli.

I realize that in step 2, there is a subtlety as to deduce your formula from Künneth's, one needs to put $Y=V= \operatorname{spec} L$ which is generally not proper over $k$. This is not really an issue as to get a Künneth formula, you only need the properness of the first term $X$.