I would like to ask for recomendation for literature on theory relative minimal models of surfaces, where "relative" in sense of that the study objects are not surfaces alone (="absolue minimal models"), but together with a dominant, flat map (="fibration") $S \to C$ to smooth curve, or more specifically to $\Bbb P^1$, so a pencil datum.
And by def, a relative minimal model $(M(S), p_M: M(S) \to C)$ of $(S, p:S \to C)$ would mean that every fibre not contains a $(-1)$-curve.
Also "surfaces" should be assumed to be proper, over alg closed base field $k$ of char $2,3$ (...to avoid troubles with elliptic pencils). Let moreover exclude the case $S \to C$ be ruled surface; as this behaves already bad in "absolute" case; indeed only in case $S$ not ruled it's birationality class contains a unique min model.
The point is that nearly all literature familar to me (Beauville, Badescu (algebraic surf.), Barth (complex s.)) discussing theory of minimal models of surfaces stick on absolute case, ie without this additional datum $(S \to C)$, but I wondering how the theory change from viewpoint of constructibility techniques to construct to a given smooth fibred surface $(S, S \to C)$ it's minimal model.
Leading problem, say we start with such $(S,S \to C)$ non ruled, how to determine it's relative minimal model $(M(S), M(S) \to C)$.
Motivation/Running problem: In the proof of Lemma 2.6 (see abs page 199) from Kondo's paper Enriques surfaces with finite automorphism groups at one stage one considered minimal model $J(Y) \to \Bbb P^1$ (...turned out to be a Jacobian fibration; ...) of an elliptic pencil $S \to \Bbb P^1$.
Then this rises basic questions about structure of this relative minimal model extractable from structure of $S$, eg there is claimed that its anticanonical linear system $\vert -K_{J(Y)} \vert$ contains a fibre, etc.
These kind of things should follow from theory of relative minimal model theory, but I don't know a source discussing this theory.
Here in comments abx proposed (assuming we have also uniqueness statement in relative situation) that the minimal model of elliptic pencil $S \to \Bbb P^1$ should be given as blowup $B$ at $9$ points of base/indeterminacy locus of rational pencil
$$ \Bbb P^2 \dashrightarrow \Bbb P^1 $$
induced by two general cubics on $\Bbb P^2$. So general fibre is elliptic and it's easy to check it not contains $(-1)$-curves in fibres, so it's minimal, especially relatively minimal.
But the raised question is why it gives actually the relative minimal model of given pencil $S \to \Bbb P^1$, so in sense that we have birational map from $S$ to it respecting both pencils to $\Bbb P^1$? Following abx's explanations this should more or less follow immediately from basics.
Going a step back, are these even bitationally equivalent? Blowup in finite # of points is clearly birationl to $\Bbb P^2$. On the other hand the function field $K(S)$ coinsides with function field of generic fibre of $S \to \Bbb P^1$, that's an elliptic curve $E$ over $K(\Bbb P^1)$, a priori I not see any reason why $E$ should be rational over base field $k$ - or, equivalently, why $K(E)=K(S)$ is purely transcendetal over $k$ in order to assure that the constructed blowup $B$ is indeed birational to it...
