have $H^1(X, K_X+p^eM)=0$ for $e\gg 0$. By Corollary \[curve-trace-cor\], $\Psi$ is a non-zero map because $\Psi$ satisfies $$\begin{aligned}
{{\operatorname{Tr}}}_C^e(M):H^0(C, K_{C}+p^eM)&\to &H^0(p^eC, K_{p^eC}+p^eM)\\
&\overset{\Psi}\to& H^0(C, K_C+M). \end{aligned}$$ Then, the trace map ${{\operatorname{Tr}}}_{X}^e(L)$ is also a non-zero map.

Surjectivity of the trace maps for surfaces ($\kappa=1$)
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In this section, we show the surjectivity of the trace map $$H^0(X, \omega_X(p^e(A+m(K_X+A))))\to H^0(X, \omega_X(A+m(K_X+A)))$$ for the case where $X$ is a surface and $\kappa(X, K_X+A)=1$. For this, we establish the following vanishing result.

\[ruled-R1\] Let $C$ be a smooth curve. Let $Y:=\mathbb P^1\times C$ and let $\pi:Y\to C$ be the projection. Let $f:X\to Y$ be the blowup at one point and let $$\theta:X\overset{f}\to Y\overset{\pi}\to C.$$ Let $A_X$ be a $\theta$-ample [$\mathbb{Z}$]{}-divisor on $X$. Then, $$R^1\theta_*(B_X^2(A_X))=0.$$

\[step-rational\] In this step, we assume $C$ is rational and we prove the assertion.

Since the assertion is local on $C$, we may assume $C\simeq \mathbb P^1.$ For an arbitrary ample [$\mathbb{Z}$]{}-divisor $A_C$ on $C$, by the Leray spectral sequence, we obtain the following exact sequence $$\begin{aligned}
0&\to & H^1(C, \theta_*(B_X^2(A_X))\otimes \mathcal O_C(A_C))\\
&\to & H^1(X, B_X^2(A_X+\theta^*A_C))\\
&\to & H^0(X, R^1\theta_*(B_X^2(A_X))\otimes \mathcal O_C(A_C))\to 0\\\end{aligned}$$ Let $A_C$ be an ample [$\mathbb{Z}$]{}-divisor on $C$ such that

1.  [$A_X+\theta^*A_C$ is ample.]{}

2.  