The maximum number of singular points on rational homology projective planes

Hwang, Dongseon; Keum, Jonghae

doi:10.1090/S1056-3911-10-00532-1

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
MR Author ID: 291447
Email: jhkeum@kias.re.kr

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).