On Weak bounded negativity conjecture

Abstract

In the first part of this article, we give bounds on self-intersections $C^2$ of integral curves $C$ on blow-ups $Bl_nX$ of surfaces $X$ with the anti-cannonical divisor $-K_X$ effective. In the last part, we prove the weak bounded negativity for self-intersections $C^2$ of integral curves $C$ in a family of surfaces $f:Y\longrightarrow B$ where $B$ is a smooth curve.