|Dept.||Tata Institute of Fundamental Research|
Smooth projective surfaces with rational Homology the same as the projective plane were an object of interest for a long time. In their seminal work Prasad and Yeung classified all such surfaces. When one allows singularities on such surfaces the problem becomes much harder. An interesting restriction (suggested by Kollar) is the existence of rational curves with only Cuspidal singularities in the smooth locus of the surface. We will discuss why the restriction is important and investigate the consequences it has for the kind of singularities the surface can admit.