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Surfaces with \(p_g = q= 1\), \(K^2 = 7\) and non-birational bicanonical maps

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Abstract

Let \(S\) be a smooth minimal projective surface of general type with \(p_g = q = 1, K_S^2 = 7\). We prove that the degree of the bicanonical map is 1 or 2. Furthermore, if the degree is 2, we describe \(S\) by a double cover.

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Acknowledgments

I am grateful to Prof. Jinxing Cai and Wenfei Liu for their valuable suggestions and many useful discussions during the preparation of this paper. Borrelli informed me that under the assumption that the bicanonical map factors through a degree 2 map, he got some results much earlier. Here I thank him for his kindness to allow me to submit this paper. I also thank the referee for his or her very valuable suggestions. The author was supported by the NSFC (No. 11226075).

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  1. School of Mathematics and Information Sciences, Shaanxi Normal University, Xi’an , 710062, People’s Republic of China

    Lei Zhang

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  1. Lei Zhang
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Zhang, L. Surfaces with \(p_g = q= 1\), \(K^2 = 7\) and non-birational bicanonical maps. Geom Dedicata 177, 293–306 (2015). https://doi.org/10.1007/s10711-014-9990-2

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  • DOI: https://doi.org/10.1007/s10711-014-9990-2

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