Let $f: X \to B$ be an elliptic fibration, iewith $B$ smooth connected curve over alg closed base field $k$, $X$ smoothsm surface, $f$ proper morphism with connected fibers such that almost all fibers are elliptic curves. Assume $f$ admits nono sections $s: B \to X$.
Then it is known that one can associate to $f: X \to B$ an up to isomorphism a unique elliptic fibration $j: J \to B$ with following properties,
called Jacobian of $f$:
(1) $J_{\eta}^{sm} \cong \operatorname{Jac}(X_{\eta})$, where $\eta$ generic point of $B$, $J_{\eta}^{sm}$ smooth locus of generic fibre, and $\operatorname{Jac}(X_{\eta}) $ Jacobian variety of curve $X_{\eta}$
(2) $J_{\eta}(K(B)) =J(B) \neq \emptyset $, ie $j$ admits sections
(3) $J^{sm}$ coinsides with Neron model of $J_{\eta}^{sm}$
The explicit construction should work sketchy like this: We take Jacobian variety $\operatorname{Jac}(X_{\eta})$ of $X_{\eta}$, form Neron model $N_J$ of it. Then there should be some projectivization of $N_J$ step be involved (...here I'm not sure why it works in detail and how uniqueness is justified at this step). Say (...modulo some magic; does anybody know a source where this step is elaborated in detail?) we succeed and obtain projective $\overline{N_J} \subset \Bbb P^n_B$, then we blow down after finitely many steps it's relative $(-1)$-curves and obtain a relative minimal model $j:J \to B$, which we would call the Jacobian of $f: X \to B$.
According to Enriques classifications of surfaces via minimal models one can associate to each smooth surfaces a bunch of birational invariant determining it's Enriques type wrt this classification;
these invariants are the plurigenera $P_n(X)=H^0(X,K_X^n)$, irregularity $q=H^1(X,O_X)$, Betti numbers $b_i(X)$ (wrt singular coho grps, if $X$ complex, otherwise wrt etale coho). Let call them all together "Enriques invariants".
Now my naive Question is what do we know about these Enriques invariants of $J$, but more precisely how are these intrinsically depend on Enriques invariants of $X \to B$? The expectation should be that as $j:J \to B$ is uniquely determined by $f:X \to B$, then Enriques invariants of $J$ are completely determined by Enriques invariants of $X$ and somehow of "choice" of fibration datum $X \to B$.
What is known (see eg. Badescu's Algebraic Surfaces, Thm 7.15) and we get it almost "for free" - in sense that it seemingly (...correct me please if I'm wrong) almost not depend on on $X \to B$ - is the statement about structure of canonical class $K_J$ of $J$, namely there is well known formula for canonical class of an elliptic fibration $g:S \to B$ as
$$ \omega_S \cong g^*(L^{-1} \otimes \omega_B) \otimes O_S(\sum_ia_iM_i) $$
where $L$ comes from decomposition $R^1g_*O_S= L \oplus T$ on smooth curve $B$, so Dedekind , with $L$ invertible sheaf and $T$ torsion sheaf on $B$ , and $M_i$ are reduced multiple fibres $g^{-1}(b_i)=m_iM_i$ of $g$.
Now in our situation the formula dramatically simplifies as $j:J \to B$ admits sections, so has no multiple fibres, so the "information" about canonical class "sits completely" in $B$.
So here we seemingly already know a lot about structure of canonical class - esp it's intersection behaviour - $K_J$ even without knowing about $X$.
But can the relation between Enriques invariants of $J$ and $X$ made be explicit / in controlled way as one may naively expect regarding $J \to B$ as canonical object associated to $f:X \to B$? So I'm looking for sources discussing the interplay between elliptic fibrations with their Jacobians from algebraic viewpoint.
Of course, one is filled with temptation above to drop the "elliptic" fibration assumtion and ask the same question for $f: X \to B$ any fibration (=$f_*O_X=O_B$) from smooth surface to smooth curve. What can one expect on Enriques invariants of associated $j:J \to B$ "extractable" from $f: X \to B$?
