Constructions of singular rational surfaces of Picard number one with ample canonical divisor

Koll'ar gave a series of examples of rational surfaces of Picard number $1$ with ample canonical divisor having cyclic singularities. In this paper, we construct several series of new examples in a geometric way, i.e., by blowing up several times inside a configuration of curves on the projective plane and then by contracting chains of rational curves. One series of our examples have the same singularities as Koll'ar’s examples.