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Surfaces with \(c_1^2 =9\) and \(\chi =5\) whose canonical classes are divisible by 3

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Abstract

We shall study minimal complex surfaces with \(c^2 = 9\) and \(\chi =5\) whose canonical classes are divisible by 3 in the integral cohomology groups, where \(c_1^2\) and \(\chi \) denote the first Chern number of an algebraic surface and the Euler characteristic of the structure sheaf, respectively. The main results are a structure theorem for such surfaces, the unirationality of the moduli space, and a description of the behavior of the canonical map. As a byproduct, we shall also rule out a certain case mentioned in a paper by Ciliberto–Francia–Mendes Lopes. Since the irregularity q vanishes for our surfaces, our surfaces have geometric genus \(p_g = 4\).

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Acknowledgements

The author expresses his gratitude to Prof. Kazuhiro Konno, who kindly gave him the comment on the normality of the canonical image (Proposition 6).

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  1. Department of Mathematics and Computer Science, Kagoshima University, Korimoto 1-21-35, Kagoshima, 890-0065, Japan

    Masaaki Murakami

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  1. Masaaki Murakami
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Correspondence to Masaaki Murakami.

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The author acknowledges the support by JSPS Grant-in-Aid Scientific Research(C) 15K04825

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Murakami, M. Surfaces with \(c_1^2 =9\) and \(\chi =5\) whose canonical classes are divisible by 3. manuscripta math. 173, 425–450 (2024). https://doi.org/10.1007/s00229-022-01442-7

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  • DOI: https://doi.org/10.1007/s00229-022-01442-7

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