Negative curves on surfaces..

The negative curve conjecture states that there is lower bound depending on a given complex projective surface $S$ for the self intersection of an effective curve $C$ on $S$. In this talk I will survey some recent work on this conjecture. By the end of the talk I will describe some work with M. Stover which shows that there is a universal constant t with the following property. The number of $C$ on $S$ having $C^2 <0$ and arithmetic genus less than $b_1(S)/4$ is bounded by $t^{r(S)-1}$, when $b_1(S)$ is the first Betti number of $S$ and $r(S)$ is the rank of the Neron Severi group of $S$.