M. Reid has a theorem as following:


Let $X$ be a non-singular projective complex surface, $L \subset \Omega^1\_X$ be a line bundle. If $h^0(L^{\otimes n})\geq2$ for some $n\geq 1$, then there is a moephism $f:X\rightarrow C$ of $X$ onto a nonsingular curve $C$ of positive genus, and a ramification divisor $D$ (of the map $f$) such that $L$ and $f^\*(\Omega^1\_C)(D)$ coincide as sheaves. 


How to derive the following statements？


(a) there exists a constant $k$ such that for all $n$ $$h^0(L^{\otimes n} )< nk$$
(b) $L$ is not in the positive cone of $NS(X)$



How to estimate the $h^0(L^{\otimes n})$
(a) The ramification divisor $D$ is contained in some fibers of $f$, so we can write $L=f^\*M(-E)$, with $M$ a line bundle on $C$ and $E$ effective. Therefore $h^0(L^{{\scriptscriptstyle\otimes} n })$ is bounded by $h^0(f^\*M^{{\scriptscriptstyle\otimes} n })=h^0(M^{{\scriptscriptstyle\otimes} n })$ (projection formula).


(b) What do you call the positive cone? We have $(L)^2\leq 0$ and $(L\cdot F)=0$ for $F$ a general fiber of $f$, so certainly $L$ is not ample. On the other hand if $D=0$ $L$ is nef. 


