Let $f: X \to B$ be an elliptic fibration, so proper map from smooth surface $X$ onto smooth conn. curve over alg closed base field $k$ with connected fibers such that almost all fibers are elliptic curves. Assume $f$ admits *no* 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 (I don't know detailed reference; is it known to somebody?) like this: We take Jacobian variety $\operatorname{Jac}(X_{\eta})$ of generic fibre $X_{\eta}$, form its Neron model $N_J$. Then there should be some projectivization of $N_J$ step be involved (...here I'm not sure how it works in detail and how uniqueness is justified at this step). Say at the end of the day (...modulo some magic; does anybody know a source where this step is elaborated in detail?) we succeed and obtain "projectivization" $\overline{N_J} \subset \Bbb P^n_B$ of $N_J$.

[...Alternatively, maybe instead one should start simpler with closed $\operatorname{Jac}(X_{\eta}) \subset \Bbb P^n_{\kappa(\eta)}$, and take it's schematic closure with respect to immersion $P^n_{\kappa(\eta)} \subset P^n_{B}$. But not sure how to reach the mentioned connection to Neron model...]

Then we proceed straight forwardly blowing down after finitely many steps all it's relative $(-1)$-curves and obtain a relative minimal model $j:J \to B$, which we would call the *Jacobian* of $f: X \to B$.  
As remarked, I would like to know a reference fixing the mentioned gaps.

Next, according to Enriques classifications of surfaces via minimal models one associates to each smooth surfaces a bunch of *birational invariants* 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](https://link.springer.com/book/10.1007/978-1-4757-3512-3), 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 $f: X \to B$?