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Let $X$ be a projective scheme over $\mathbb{C}$. For a sheaf on $X$, $ p_E(d)=\chi(X,E(d)) $ be the Hilbert polynomial of $E$. A sheaf $E$ on $X$ is siad to be stable if for every proper subsheaf $F\subset E$, $ p_F(d)/rk(F) for sufficiently large $d>0$. Why is any rank $1$ sheaf always stable?

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    @QiL You are right. Thank you for pointing this out.2012-11-12

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If $F$ is a non-zero sub-coherent sheaf of $E$, then it also has rank 1. Let $S=E/F$. Then $p_E(d)=p_F(d)+p_S(d)$ with $p_S(d)>0$ when $d$ is big enough. Thus $p_F(d)/\mathrm{rk}(F)=p_F(d) < p_E(d)=p_E(d)/\mathrm{rk}(E).$