Construction of singular rational surfaces of Picard number one with ample canonical divisor (Q2884408)

The authors give geometric constructions of singular rational surfaces of Picard number one and ample canonical class. Prior examples of such surfaces were given by \textit{J. Kollár} [Pure Appl. Math. Q. 4, No. 2, 203--236 (2008; Zbl 1145.14031)] and \textit{S. Keel} and \textit{J. McKernan} [``Rational curves on quasi-projective surfaces'', Mem. Am. Math. Soc. 669, 153 p. (1999; Zbl 0955.14031)]. The previously known examples all contain two quotient singularities. The constructions in this paper give surfaces with up to three cyclic singularities.NEWLINENEWLINETheir constructions involve surfaces with the same Betti numbers as the complex projective plane. Such surfaces are called \(\mathbb{Q}\)-homology projective planes. In an earlier paper, the authors showed that these surfaces may have up to four quotient singularities. Their constructions create examples of rational \(\mathbb{Q}\)-homology projective planes with up to three quotient singularities. It is conjectured that this is the maximal number of quotient singularities such a surface can have.NEWLINENEWLINETheir first construction follows the work of Kollár. They consider a class of hypersurfaces in weighted projective space then blow down two rational curves to obtain quotient singularities. The remaining constructions involve blowing up points and contracting chains of rational curves.