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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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  • $\begingroup$ Well, if $M^2>0$, the Hodge index theorem gives $$ K^2\le \frac{(K.M)^2}{M^2}\le (K.M)^2 $$ $\endgroup$
    – J.C. Ottem
    Commented Jan 22, 2011 at 1:18
  • $\begingroup$ Yes, right. But as $p_g$ goes large, this bound seems not so beautiful. I do not know if there is a linear bound. $\endgroup$
    – Tong
    Commented Jan 22, 2011 at 1:36
  • $\begingroup$ If the canonical image of $S$ is a surface, then $M^2\ge p_g-1$. This improves the bound suggested by JC. $\endgroup$
    – rita
    Commented Jan 22, 2011 at 17:01
  • $\begingroup$ @rita: right. But I think using the technique of the proof of Noether inequality, maybe $M^2 \ge 2p_g-4$. $\endgroup$
    – Tong
    Commented Jan 22, 2011 at 18:24
  • $\begingroup$ @Michael: that's right. $\endgroup$
    – rita
    Commented Jan 23, 2011 at 17:14

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