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, $K_SM \ge aK_S^2$?

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.