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Monodromy representations and surfaces with maximal Albanese dimension

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More and more I’m aware that the permutations are not unlimited.

Russell Hoban, Turtle Diary

Abstract

We relate the existence of some surfaces of general type and maximal Albanese dimension to the existence of some monodromy representations of the braid group \(\mathsf {B}_2(C_2)\) in the symmetric group \(\mathsf {S}_n\). Furthermore, we compute the number of such representations up to \(n=9\), and we analyze the cases \(n \in \{2, \, 3, \, 4\}\). For \(n=2, \, 3\) we recover some surfaces with \(p_g=q=2\) recently studied (with different methods) by the author and his collaborators, whereas for \(n=4\) we obtain some conjecturally new examples.

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Acknowledgements

The author was partially supported by GNSAGA-INdAM. He thanks the organizers of the workshop Birational Geometry of Surfaces (University of Rome Tor Vergata, January 2016) for the invitation and the hospitality. He is also indebted with “abx”, “aglearner”, Ariyan Javanpeykar, Stefan Behrens and Mohan Ramachandran for interesting discussions on several MathOverflow threads, with R. Pardini and V. Coti Zelati for their support during the editing process and with the anonymous referee for helpful comments and remarks.

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  1. Dipartimento di Matematica e Informatica, Università della Calabria, Cubo 30B, 87036, Arcavacata di Rende, Cosenza, Italy

    Francesco Polizzi

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  1. Francesco Polizzi
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Correspondence to Francesco Polizzi.

Appendix: the GAP4 script

Appendix: the GAP4 script

This short appendix contains the GAP4 script used in the paper. We explicitly write down the version for \(n=3\). For the other cases, it suffices to change the first line n=3 to the desired value of n.

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Polizzi, F. Monodromy representations and surfaces with maximal Albanese dimension. Boll Unione Mat Ital 11, 107–119 (2018). https://doi.org/10.1007/s40574-017-0131-3

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