I'm reading a proof of del Busto (<http://arxiv.org/pdf/alg-geom/9410018v1.pdf>) and am confused at one step. I'm sure its quite easy, but the key trick hasn't come to me yet.


The context here is that $X$ is a nonsingular (complex) projective surface and A an ample divisor. For $\kappa\_1,\ldots,\kappa\_r$ positive integers $\{x\_1,\ldots,x\_r\}$ a collection of $r$ distinct points on $X$ and $Z = \sum \kappa\_i x\_i$, let $f \colon Y \to X$ be the blow up of $X$ at $\{x\_1,\ldots,x\_r\}$. Denote by $E\_1,\ldots,E\_r \subset Y$ the corresponding exceptional divisors. He reduces the problem to the vanishing


\begin{equation} V(k) := H^1\left(Y, \mathcal{O}\_Y \left( f^\*(kA) - \sum\_{i=1}^r \kappa\_i E\_i \right)\right) = 0. \end{equation}


Set $B = f^\*(kA - K\_X) - \sum\_{i=1}^r (\kappa\_i + 1)E\_i$. For $n > 0$ set $nB = M\_n + F\_n$ its Zariski decomposition. For $k$ larger than his bound, $\frac{1}{n} F\_n \cdot f^\* A < 1$ so we can write $\lfloor \frac{1}{n} F\_n \rfloor = \sum\_{i=1}^r \eta\_i E\_i$ with $\eta\_i \geq 0$. He next shows as a consequence of Kawamata-Viehweg that


\begin{equation} W(k) := H^1\left(Y, f^\* \mathcal{O}\_X\left( (k+1) A \right) \otimes \mathcal{O}\_Y \left( - \sum\_{i=1}^r (\kappa\_i + \eta\_i) E\_i \right) \right) = 0\end{equation} and deduces the result from the claimed implication $W(k) = 0 \Rightarrow V(k+1) = 0$. The two sheaves are very similar, in particular off by $- \sum\_{i=1}^r \eta\_i E\_i $. However, I don't easily see this implication. 



A key step in Del Busto's effective Matsusaka theorem
It seems to me that $[\frac 1nF\_n]$ is exc. and there is a short exact sequence inducing $$W(k)→V(k+1)→H^1((−∑k\_iE\_i)|\_{\sum \eta \_iE\_i})$$ which has positive degree on a bunch of (non-reduced $\mathbb P^1$'s and so vanishes. If each $\eta \_i\in \{0,1\}$, this is clear. If not proceed by induction via the short exact sequences of the form $$0\to (-k-(\eta-1))E)|\_E\to (-kE)|\_{\eta E}\to (-kE)|\_{(\eta -1)E}\to 0.$$ 


