The bounded negativity conjecture

By a negative curve on a smooth projective surface we mean a reduced, irreducible curve $C$ with $C^2<0$. The bounded negativity conjecture. For each smooth complex projective surface $X$ there exists a number $b(X)\ge 0$ such that

$$C^2\ge -b(X)$$

for every negative curve $C\subseteq X$. This conjecture asks for a uniform lower bound on the self-intersection numbers of negative curves on each fixed surface and is a central problem in the theory of projective surfaces. Its status is not specified in the source.