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?

## 1 Answer

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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.