On homology projective planes

Abstract

We study extremal objects among singular algebraic surfaces: rational homology projective planes, i.e., normal projective surfaces which have the same Betti numbers as the complex projective plane. The number of singular points on rational homology projective planes is not bounded in general. However, it is known that if we allow only quotient singularities, then the number of singular points is bounded above by 5. There are many examples with 4 singularities, but no examples with 5 singularities are known. As a main result of this thesis, we determine the maximum number of singular points on rational homology projective planes with quotient singularities. More precisely, we prove that if a rational homology projective plane with quotient singularities has exactly 5 singular points, then it has singularities of type $3A_1 \oplus 2A_3$, and its minimal resolution is a smooth Enriques surface. An example of a rational homology projective plane with $5$ quotient singularities can be constructed by contracting suitable 9 curves forming $3A_1 \oplus 2A_3$ on an Enriques surface. We also prove some intersection-theoretic formulas for curves on the minimal resolution of rational homology projective planes with quotient singularities, and a number-theoretic formula on Hirzebruch-Jung continued fractions.