I've been going through a bit of the literature on the classification of Fano varieties, and it was pointed out to me by one of my committee members that I overlooked an important question regarding the existence of smooth surface sections which doesn't follow from Bertini's theorem.
Let $X$ be a prime Fano threefold of index $1$ (that is to say $\operatorname{Pic}(X) = \mathbb{Z} H$ with $H = -K_X$). When $\operatorname{char} k = 0$, one can use Kodaira vanishing and Grothendieck-Riemann-Roch to show $\dim_k |H| \geq 1$, so there always exists an anticanonical section $S \in |H|$ (alternatively, Shokurov used a clever way of rewriting the Hilbert polynomial, as discussed in Debarre's notes on classifying Fano varieties). Under the additional assumption that $S$ is smooth, one may use adjunction and a quick computation of $h^1(\mathcal{O}_S)$ to show $S$ is K3 (thus leading to the desired rank 2 vector bundle in Mukai's classification). However, as pointed out in the footnote on p.6 of Debarre's notes linked above, Mukai assumed a general member of $|H|$ to be smooth, which need not be true in general.
While the requirement that $\operatorname{char} k = 0$ can be circumvented, as explained in Proposition 6.3 of Bayer, Lahoz, Macri, Stellari 2022, it seems there is still an assumption of generic smoothness in constructing a K3 fibration that I am now not seeing why it should exist.
Any insight into how this detail is resolved would be very much appreciated.
