I had been reading a couple of texts by J.P. Demailly, one of them titled "Effective bounds for very ample line bundles". In the introduction the author mentions a result due to I. Reider (stated in "Vector bundles of rank $2$ and linear systems on algebraic surfaces") which states that for a smooth projective complex surface $X$, if $L$ is an ample line bundle then $K\_X+4L$ is very ample. I had been reading the article by Reider but am not able to find this result. Could some one tell me what I am missing?



A question on very ample line bundle on smooth projective surfaces
It is not stated as such in Reider's paper, but it is an easy consequence of his main theorem (thm. 1) : Reider proves that if $N$ is nef and $N^2\geq 10$, $K\_X+N$ is very ample unless there is an effective divisor $E$ on $X$ with $E.N\leq 2$ (+ some extra conditions). If $N=4L\ $ with $L$ ample, clearly such a divisor cannot exist, and $\ N^2\geq 16$, q.e.d.


