Weak Bounded Negativity Conjecture

In this paper, we prove the following “Weak Bounded Negativity Conjecture”, which says that given a complex smooth projective surface $X$, for any reduced curve $C$ in $X$ and integer $g$, assume that the geometric genus of each component of $C$ is bounded from above by $g$, then the self-intersection number $C^2$ is bounded from below.