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.

- Description
- 한국과학기술원 : 수리과학과,

- Publisher
- 한국과학기술원

- Issue Date
- 2009

- Identifier
- 327742/325007 / 020037685

- Language
- eng

- Description
한국과학기술원 : 수리과학과, 한국과학기술원 : 수리과학과, 2009. 8., [ iii, 65 p. ]

- Keywords
rational homology projective plane; quotient singularity; orbifold Bogomolov-Miyaoka-Yau inequality; integral quadratic form; (-1)-curve; 유리 호몰로지 사영평면; 상 특이점; 오비폴드 보고몰로프-미야오카-야우 부등식; 정수계수 이차형식; (-1)-곡선; rational homology projective plane; quotient singularity; orbifold Bogomolov-Miyaoka-Yau inequality; integral quadratic form; (-1)-curve; 유리 호몰로지 사영평면; 상 특이점; 오비폴드 보고몰로프-미야오카-야우 부등식; 정수계수 이차형식; (-1)-곡선

- Appears in Collection
- MA-Theses_Ph.D.(박사논문)

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