The negative curve conjecture states that there is lower
bound depending on a given complex projective surface for the
self intersection of an effective curve on . 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 on having
and arithmetic genus less than is bounded by ,
when is the first Betti number of and is the rank
of the Neron Severi group of .