Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

The maximum number of singular points on rational homology projective planes


Authors: Dongseon Hwang and Jonghae Keum
Journal: J. Algebraic Geom. 20 (2011), 495-523
DOI: https://doi.org/10.1090/S1056-3911-10-00532-1
Published electronically: March 24, 2010
MathSciNet review: 2786664
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Abstract | References | Additional Information

Abstract: A normal projective complex surface is called a rational homology projective plane if it has the same Betti numbers with the complex projective plane $ \mathbb{C}\mathbb{P}^2$. It is known that a rational homology projective plane with quotient singularities has at most $ 5$ singular points. So far all known examples have at most $ 4$ singular points. In this paper, we prove that a rational homology projective plane $ S$ with quotient singularities such that $ K_S$ is nef has at most $ 4$ singular points except one case. The exceptional case comes from Enriques surfaces with a configuration of 9 smooth rational curves whose Dynkin diagram is of type $ 3A_1 \oplus 2A_3$.

We also obtain a similar result in the differentiable case and in the symplectic case under certain assumptions which all hold in the algebraic case.


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Additional Information

Dongseon Hwang
Affiliation: Department of Mathematics, Korea Advanced Institute of Science and Technology, Daejon #305-701, Korea
Address at time of publication: School of Mathematics, Korea Institute for Advanced Study, Seoul 130-722, Korea
Email: themiso@kaist.ac.kr, dshwang@kias.re.kr

Jonghae Keum
Affiliation: School of Mathematics, Korea Institute For Advanced Study, Seoul 130-722, Korea
Email: jhkeum@kias.re.kr

DOI: https://doi.org/10.1090/S1056-3911-10-00532-1
Received by editor(s): September 12, 2008
Received by editor(s) in revised form: February 9, 2009
Published electronically: March 24, 2010
Additional Notes: Research supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2009-0063180).

American Mathematical Society