Let $f: X \to B$ be an elliptic fibration, ie $B$ smooth connected curve over alg closed base field $k$, $X$ smooth surface, $f$  proper morphism 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 unique elliptic fibration $j: J \to B$ with following properties:

(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 $\operatorname{Jac}(X_{\eta})$. Then there should be some projectivation 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 $\overline{J} \subset \Bbb P^n_B$, 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 $f: X \to B$.

Now 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 $X \to B$.


Now what is known and we get 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 canonical class $K_J$ of $J$, namely there is well known formula for kanonical 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 $R^1g_*O_S= L \oplus T$, with $L$ invertible sheaf and $T$ torsion sheaf on $B$ , and $M_i$ are reduced multiple fibres $F_i=m_iP_i$ 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 $f:X \to B$?

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 $f: X \to B$?