show $$R^1\theta_*(B_X^2(A'))=0$$ for every ample [$\mathbb{Z}$]{}-divisor $A'$.

By Remark \[cartier-remark\] and Remark \[cartier-trace\], we obtain the following exact sequence $$0\to B_X^2\to F_*\omega_X\overset{{{\operatorname{Tr}}}_X^1}\to \omega_X\to 0.$$ By Lemma \[e-vs-e+1\], it is sufficient to show that $$H^1(X, B_X^2(p^d(A+m(K_X+A))))=0$$ for every $0\leq d\leq e$. We can write $K_X+A=\theta^*H$ where $H$ is an ample [$\mathbb{Z}$]{}-divisor on $C$. By the Leray spectral sequence, we obtain $$\begin{aligned}
0&\to&H^1(C, \theta_*(B_X^2(p^d(A+m(K_X+A)))))=0\\
&\to&H^1(X, B_X^2(p^d(A+m(K_X+A))))\\
&\to&H^0(C, R^1\theta_*(B_X^2(p^d(A+m(K_X+A))))\to 0.\end{aligned}$$ The first term vanishes by the Serre vanishing theorem. Thus, it is sufficient to show that $R^1\theta_*(B_X^2(A'))=0$ for every ample [$\mathbb{Z}$]{}-divisor $A'$.

In this step, we prove the assertion. By Step \[step-R1\], it is sufficient to show $$R^1\theta_*(B_X^2(A'))=0$$ for an ample [$\mathbb{Z}$]{}-divisor $A'$. Let $$\theta:X\overset{f}\to Y\overset{\pi}\to C$$ where $\pi$ is a $\mathbb P^1$-bundle structure. Since the problem is local on $C$, we may assume that $\theta$ has only one singular fiber and that $Y=\mathbb P^1\times C.$ Let $F_s$ be the singular fiber. Then, we see $$0=(K_X+A)\cdot F_s=-2+A\cdot F_s.$$ Thus, $F_s$ has at most two irreducible components. This implies that $f$ is the blowup at one point. Then, the assertion follows from Proposition \[ruled-R1\].

Surjectivity of the trace maps for surfaces ($\kappa=2$)
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In this section, we show the surjectivity of the trace map 