$$H^0(X, \omega_X(p^e(A+m(K_X+A))))\to H^0(X, \omega_X(A+m(K_X+A)))$$ for the case where $X$ is a surface and $\kappa(X, K_X+A)=2$. Let us recall a lemma on the global generation.

\[uniform-generation\] Let $X$ be a smooth projective variety. Let $A$ be an ample [$\mathbb{Z}$]{}-divisor and let $G$ be a coherent sheaf. Then, there exists $n_0\in\mathbb Z_{>0}$, depending only on $A$ and $G$, such that $$G(n_0A+N)$$ is generated by global sections for every nef [$\mathbb{Z}$]{}-divisor $N$.

The assertion immediately follows from the Castelnuovo–Mumford regularity ([@Lazarsfeld Theorem 1.8.5]) and the Fujita vanishing theorem ([@Fujita1 Theorem (1)], [@Fujita2 Section 5]).

To prove the surjectivity, we establish the following vanishing result.

\[birat-B-vanish\] Let $h:X\to Z$ be a birational morphism between smooth projective surfaces. Let $A_X$ be an ample [$\mathbb{Z}$]{}-divisor and let $A_Z$ be an ample [$\mathbb{Z}$]{}-divisor on $Z$. Then, there exists $m_0\in \mathbb Z_{>0}$ such that $$H^1(X, B_X^2(A_X+h^*(m_0A_Z+N_Z)))=0$$ for every nef [$\mathbb{Z}$]{}-divisor $N_Z$ on $Z$.

The birational morphism $h$ is an $n$-times blowups. We show the assertion by the induction on $n$.

\[step-n=0\] If $n=0$, then the assertion follows from the Fujita vanishing theorem ([@Fujita1 Theorem (1)], [@Fujita2 Section 5]). Thus, we may assume that $n>0$ and the assertion holds for $n-1$.

\[step-one-point\] In this step, we prove that we 