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Towards generic base-point-freeness for hyperkähler manifolds of generalized Kummer type

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Abstract

We study base-point-freeness for big and nef line bundles on hyperkähler manifolds of generalized Kummer type: For \(n\in \{2,3,4\}\), we show that, generically in all but a finite number of irreducible components of the moduli space of polarized \(\textrm{Kum}^n\)-type varieties, the polarization is base-point-free. We also prove generic base-point-freeness in the moduli space in all dimensions if the polarization has divisibility one.

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Acknowledgements

This work is part of my Master’s thesis at ETH Zürich. I would like to thank Ulrike Rieß  for suggesting me this topic, for supervising my thesis, and for her continuous advice and support. I am also grateful for the financial support provided by ETH Foundation during my master’s, and by ERC Synergy Grant HyperK (Grant agreement No. 854361) during the completion and editing of the article. In particular, I express my gratitude to Daniele Agostini, Fabrizio Anella, and Daniel Huybrechts for reading a preliminary version of this paper.

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  1. Mathematisches Institut, Universitat Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Mauro Varesco

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  1. Mauro Varesco
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Correspondence to Mauro Varesco.

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Communicated by Daniel Greb.

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Varesco, M. Towards generic base-point-freeness for hyperkähler manifolds of generalized Kummer type. Abh. Math. Semin. Univ. Hambg. 93, 133–147 (2023). https://doi.org/10.1007/s12188-023-00271-z

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