Abstract
We construct a simply connected minimal complex surface of general type with \(p_g=0\) and \(K^2=2\) which has an involution such that the minimal resolution of the quotient by the involution is a simply connected minimal complex surface of general type with \(p_g=0\) and \(K^2=1\). In order to construct the example, we combine a double covering and \(\mathbb Q \)-Gorenstein deformation. Especially, we develop a method for proving unobstructedness for deformations of a singular surface by generalizing a result of Burns and Wahl which characterizes the space of first order deformations of a singular surface with only rational double points. We describe the stable model in the sense of Kollár and Shepherd-Barron of the singular surfaces used for constructing the example. We count the dimension of the invariant part of the deformation space of the example under the induced \(\mathbb Z /{2}\mathbb Z \)-action.
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Acknowledgments
The authors would like to thank Professor Yongnam Lee for helpful discussion during the work, careful reading of the draft version, and many valuable comments. The authors also wish to thank Professor Jenia Tevelev for indicating a mistake in an earlier version of this paper, and the referee especially for the remark on the proof of Proposition 3.7 which makes it simpler. Heesang park was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) grant funded by the Korean Government (2011-0012111). Dongsoo Shin was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (2010-0002678). Giancarlo Urzúa was supported by a FONDECYT Inicio grant funded by the Chilean Government (11110047).
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Park, H., Shin, D. & Urzúa, G. A simply connected numerical Campedelli surface with an involution. Math. Ann. 357, 31–49 (2013). https://doi.org/10.1007/s00208-013-0905-6
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DOI: https://doi.org/10.1007/s00208-013-0905-6