first equation follows from Lemma \[cartier-etale\]. The last equation follows from Step \[step-rational\] and Step \[step-H1-R1\]. Then, the assertion holds by Step \[step-H1-R1\].

Let us prove the main theorem in this section.

\[trace-surface-1\] Let $X$ be a smooth projective surface and let $A$ be an ample [$\mathbb{Z}$]{}-divisor such that $\kappa(X, K_X+A)=1$ and that $K_X+A$ is nef. Then, there exists $m_1\in\mathbb Z_{>0}$ such that the trace map ${{\operatorname{Tr}}}_X^e(A+m(K_X+A))$ $$H^0(X, \omega_X(p^e(A+m(K_X+A))))\to H^0(X, \omega_X(A+m(K_X+A)))$$ is surjective for every $m\geq m_1$ and for every $e\in\mathbb Z_{>0}$.

We see that $K_X+A$ is semi-ample by [@Fujita3].

In this step, we prove that, for some $n_0\in\mathbb Z_{>0}$, the complete linear system $$\Phi_{|n_0(K_X+A)|}=:\theta:X\to C$$ gives a ruled surface structure, that is, $\theta$ is a projective morphism to a smooth projective curve such that $\theta_*\mathcal O_X=\mathcal O_C$ and that a general fiber is $\mathbb P^1$.

We can find $n_0\in\mathbb Z_{>0}$ such that $$\Phi_{|n_0(K_X+A)|}=:\theta:X\to C$$ is a projective morphism to a smooth projective curve such that $\theta_*\mathcal O_X=\mathcal O_C$. Then, by [@Badescu Corollary 7.3], general fibers are integral. Since a general fiber $F$ satisfies $$0=(K_X+A)\cdot F>(K_X+F)\cdot F,$$ we see $F\simeq \mathbb P^1.$ Thus, $\theta$ gives a ruled surface structure.

\[step-R1\] In this step, we prove that it is sufficient to 