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Koll\'ar gave a series of examples of rational surfaces of Picard number $1$with ample canonical divisor having cyclic singularities. In this paper, weconstruct several series of new examples in a geometric way, i.e., by blowingup several times inside a configuration of curves on the projective plane andthen by contracting chains of rational curves. One series of our examples havethe same singularities as Koll\'ar's examples.

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