Suppose that $X$ is a smooth algebraic variety over an algebraically closed (uncountable if it helps) field of characteristic $p > 0$. Suppose that $L$ is a line bundle, probably ample or at least positive, and that $\delta \subseteq |H^0(X, L)|$ is a linear system.
It is well known that just because $\delta$ is base-point-free, it does not mean that a general member defines a smooth subscheme (it need not even be reduced, for example Frobenius pull-backs of linear systems, Remark 10.9.3 in Hartshorne's algebraic geometry).
However, let's suppose the following:
$f : X \to Y$ is a map, $M$ is a very ample line bundle on $Y$ and $L = f^* M$. Suppose further that $\delta$ is the pull-back of the complete linear system $|H^0(Y, M)|$. Are there any (separability?) conditions on the map $f$ which still guarantee that Bertini holds for $\delta$? I imagine this must be well known, but I don't know the right references.
In particular, I am looking for conditions weaker than etale (/ etale outside a finite set of points)? Say in the birational case, or the finite case?
Best Answer
I think Corollary 4.3 of Spreafico's Axiomatic theory for transversality and Bertini type theorems does what you want. It says (in the case where the property is taken to be smoothness) that if $f:X\to \mathbb P^n$ is a finite type morphism from a smooth scheme $X$ over any infinite field, and if $f$ is residually separated (i.e. the induced extensions of residue fields are separable), then the pullback of a generic hyperplane is smooth.