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Construction and characterisation of the varieties of the third row of the Freudenthal–Tits magic square

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Abstract

We characterise the varieties appearing in the third row of the Freudenthal–Tits magic square over an arbitrary field, in both the split and non-split version, as originally presented by Jacques Tits in his Habilitation thesis. In particular, we characterise the variety related to the 56-dimensional module of a Chevalley group of exceptional type \(\mathsf {E_7}\) over an arbitrary field. We use an elementary axiom system which is the natural continuation of the one characterising the varieties of the second row of the magic square. We provide an explicit common construction of all characterised varieties as the quadratic Zariski closure of the image of a newly defined affine dual polar Veronese map. We also provide a construction of each of these varieties as the common null set of quadratic forms.

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Acknowledgements

The research leading to the first part of this paper was carried out in Auckland while the first and third author were trapped in their accommodation by lockdown due to COVID-19, still graciously enjoying the help and hospitality of the second author. Our gratitude also goes to the referee for their willingness and courage to read in detail through this long paper.

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Authors and Affiliations

  1. Department of Mathematics, Ghent University, Ghent, Belgium

    Anneleen De Schepper, Hendrik Van Maldeghem & Magali Victoor

  2. Department of Mathematics, University of Auckland (UoA), Auckland, New Zealand

    Jeroen Schillewaert

Authors
  1. Anneleen De Schepper
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  2. Jeroen Schillewaert
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  3. Hendrik Van Maldeghem
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  4. Magali Victoor
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Correspondence to Anneleen De Schepper.

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Anneleen De Schepper: supported by the Fund for Scientific Research Flanders—FWO Vlaanderen. Jeroen Schillewaert: supported by UoA FRDF grant and by New Zealand Marsden Fund grant MFP-UOA-2122.

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De Schepper, A., Schillewaert, J., Van Maldeghem, H. et al. Construction and characterisation of the varieties of the third row of the Freudenthal–Tits magic square. Geom Dedicata 218, 20 (2024). https://doi.org/10.1007/s10711-023-00864-1

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  • DOI: https://doi.org/10.1007/s10711-023-00864-1

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