@naf: Thanks but sorry can you elaborate? Why does your comment imply that such an S does not exist?

I see. Sorry can you elaborate? By ampleness and Mori, the line bundle of every surface is presumably ample and hence no (r,0) ones I suppose. But why are there (1,0) curves? Sorry if I am making a stupid mistake but the ample hypersurface is CP3 and for that this property appears to be true.

I guess I am asking whether given a C, there is an S, such that a multiple of C is S cupped with omega.