Jacobian fibration of elliptic fibration: basic relations between Enriques invariants

Question

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, 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 $f: X \to B$?

Answer 1

Firstly, note that we cannot expect the relationship between the plurigenera of $X$ and those of $J$ to be particularly easy. For example, passing from $X$ to $J$ can change the Kodaira dimension: If $X$ is an Enriques surface, then for any choice of an elliptic fibration on $X$, its Jacobian $J$ is a rational surface. This is because the elliptic fibration on an Enriques surface always has multiple fibers and passing from $X$ to $J$ gets rid of them. But in some sense, multiple fibers are really the main culprit here. In particular, the Kodaira dimension does not change if $X \to B$ does not have multiple fibers. Moreover, even in the presence of multiple fibers, we always have $\kappa(X) \geq \kappa(J)$.

Despite this, we can quite accurately say what the Enriques invariants of $J$ are, given only the Enriques invariants of $X$:

For the Betti numbers we always have $b_i(X) = b_i(J)$. This is Corollary 5.3.5 in the book "Enriques Surfaces I" by Cossec and Dolgachev. In general, Chapter 5 of that book is a good reference for these kinds of things.

For the plurigenera, let us first look at what happens in the Jacobian case. If $J$ is a Jacobian elliptic surface over the base curve $B$ of genus $g_B$, the canonical bundle of $J$ is given as $2g_B - 2 + \chi(\mathcal{O}_J)$ times the fiber class (at least up to algebraic equivalence). Consequently, we see that $\omega_J$ is the pullback of a line bundle $\mathcal{L}$ on $B$ of degree $2g_B - 2 + \chi(\mathcal{O}_J)$. With this notation, we have that $P_n(J) = h^0(\mathcal{L}^{\otimes n})$ and the latter we can compute by Riemann-Roch. Thus, if $\mathcal{L}$ has positive degree and $J$ is not a product ($J$ is a product iff $\mathcal{L} \cong \omega_B$), we have $P_n(J) = n(2g_B-2+\chi(\mathcal{O}_J))+1-g_B$. If $\mathcal{L}$ has negative degree, we have $P_n(J) = 0$ for all $n$. The degree zero case can be dealt with using the classification of surfaces; essentially we see that either $J$ is either a K3 surface or an abelian surface or a bielliptic surface. (We have $g_B = 0$ in the first case and $g_B = 1$ in the other two. Moreover, $J$ being an abelian surface happens only if it is a product. This is because $J \to B$ has a section, so that we can form the quotient $J/B$ as an abelian variety; this yields a morphism $J \to B \times (J/B)$ which is injective, hence an isomorphism.). In any case, the plurigenera of these are well-known.

The point of all this was that the previous paragraph shows that we can essentially recover the plurigenera $P_n(J)$ simply from knowing $\chi(\mathcal{O}_J)$ and $g_B$. Now, $g_B$ clearly does not change when passing from $X$ to $J$ and neither does $\chi(\mathcal{O}_X)$ (this is in Chapter 5 of Cossec-Dolgachev). As $\chi(\mathcal{O}_X) = 1 - q(X) + P_1(X)$ by Serre duality, this shows that (modulo some hiccups regarding the case when $J$ is a product) we can completely recover the $P_n(J)$ given only $q(X)$ and $P_n(X)$ (unless $J$ is a product, we have $g_B = q(X)$).

Edit: To justify the claims in the first paragraph, note that even if $X \to B$ is not Jacobian, we have an explicit formula for the canonical class of $X$: As a $\mathbb{Q}$-divisor, it is given as $2g_B - 2 + \chi(\mathcal{O}_X) + \sum_{i} (1-\frac{1}{m_i})$ times the fiber class, where the sum runs over the multiple fibers of $X \to B$. In particular, if $m$ is the least common multiple of the $m_i$, we see that $\omega_X^{\otimes m}$ is again the pullback of a line bundle on $B$, which is of degree $m(2g_B - 2 + \chi(\mathcal{O}_X) + \sum_{i} (1-\frac{1}{m_i}))$. Hence the Kodaira dimension of $X$ is $-\infty$, $0$ or $1$ depending on whether this degree is negative, zero, or positive respectively. Passing from $X$ to $J$ only removes the contribution of the multiple fibers, so the Kodaira dimension can only decrease if there are multiple fibers.