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Let $S$ be a minimal surface of general type over $\mathbb{C}$ with $p_g=h^0(K_S)>1$. As a convention, we can write $|K_S|=|M|+F$ such that $F$ is the fixed part. We know that $K_SM \le K_S^2$.

However, is there a lower bound of $K_SM$ given by $K_S^2$? For example, $K_SM \ge aK_S^2$, which means that $K_SF$ can not be too big.

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Well, if $M^2>0$, the Hodge index theorem gives $$ K^2\le \frac{(K.M)^2}{M^2}\le (K.M)^2 $$ – J.C. Ottem Jan 22 '11 at 1:18
Yes, right. But as $p_g$ goes large, this bound seems not so beautiful. I do not know if there is a linear bound. – Tong Jan 22 '11 at 1:36
If the canonical image of $S$ is a surface, then $M^2\ge p_g-1$. This improves the bound suggested by JC. – rita Jan 22 '11 at 17:01
@rita: right. But I think using the technique of the proof of Noether inequality, maybe $M^2 \ge 2p_g-4$. – Tong Jan 22 '11 at 18:24
@Michael: that's right. – rita Jan 23 '11 at 17:14

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