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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.

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). 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 $g_B = q(X)$).

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.

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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). 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 $g_B = q(X)$).

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 it is a product). 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 $g_B = q(X)$).

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). 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 $g_B = q(X)$).

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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}_X))+1-g_B$$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 it is a product). 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 $g_B = q(X)$).

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 $P_n(J) = n(2g_B-2+\chi(\mathcal{O}_X))+1-g_B$. 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 it is a product). 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 $g_B = q(X)$).

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 it is a product). 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 $g_B = q(X)$).

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