# Algebraic surfaces of general type with \(p_g=q=1\) and genus 2 Albanese fibrations

## Abstract

In this paper, we study algebraic surfaces of general type with \(p_g=q=1\) and genus 2 Albanese fibrations. We first study the examples of surfaces with \(p_g=q=1, K^2=5\) and genus 2 Albanese fibrations constructed by Catanese using singular bidouble covers of \(\mathbb {P}^2\). We prove that these surfaces give an irreducible and connected component of \(\mathcal {M}_{1,1}^{5,2}\), the Gieseker moduli space of surfaces of general type with \(p_g=q=1, K^2=5\) and genus 2 Albanese fibrations. Then by constructing surfaces with \(p_g=q=1,K^2=3\) and a genus 2 Albanese fibration such that the number of the summands of the direct image of the bicanonical sheaf (under the Albanese map) is 2, we give a negative answer to a question of Pignatelli.

## Notes

### Acknowledgements

The author is sponsored by China Scholarship Council “High-level university graduate program”. The author would like to thank his advisor, Professor Fabrizio Catanese at Universität Bayreuth, for suggesting this research topic, for a lot of inspiring discussion with the author and for his encouragement to the author. The author would also like to thank his domestic advisor, Professor Jinxing Cai at Beijing University, for his encouragement and some useful suggestions. The author is grateful to Binru Li and Roberto Pignatelli for a lot of helpful discussion.

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