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
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).