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A uniform characterisation of the varieties of the second row of the Freudenthal-Tits magic square over arbitrary fields

Anneleen De Schepper, Jeroen Schillewaert (UGent) and Hendrik Van Maldeghem (UGent)
(2023) JOURNAL OF COMBINATORIAL ALGEBRA. 7(3-4). p.227-282
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Abstract
We characterise the projective varieties related to the second row of the Freudenthal- Tits magic square, for both the split and the non-split form, using a common, simple and short geometric axiom system. A special case of our result simultaneously captures the analogues over arbitrary fields of the Severi varieties (comprising the 27-dimensional E6 module and some of its subvarieties), as well as the Veronese representations of projective planes over composition division algebras (most notably the Cayley plane). It is the culmination of almost four decades of work since the original 1984 result by Mazzocca and Melone who characterised the quadric Veronese variety over a finite field of odd order. The latter result is a finite counterpart to the characterisation of the complex quadric Veronese surface by Severi from 1901.
Keywords
Freudenthal-Tits magic square, Cartan variety, Veronese representation, COMBINATORIAL

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MLA
De Schepper, Anneleen, et al. “A Uniform Characterisation of the Varieties of the Second Row of the Freudenthal-Tits Magic Square over Arbitrary Fields.” JOURNAL OF COMBINATORIAL ALGEBRA, vol. 7, no. 3–4, 2023, pp. 227–82, doi:10.4171/JCA/75.
APA
De Schepper, A., Schillewaert, J., & Van Maldeghem, H. (2023). A uniform characterisation of the varieties of the second row of the Freudenthal-Tits magic square over arbitrary fields. JOURNAL OF COMBINATORIAL ALGEBRA, 7(3–4), 227–282. https://doi.org/10.4171/JCA/75
Chicago author-date
De Schepper, Anneleen, Jeroen Schillewaert, and Hendrik Van Maldeghem. 2023. “A Uniform Characterisation of the Varieties of the Second Row of the Freudenthal-Tits Magic Square over Arbitrary Fields.” JOURNAL OF COMBINATORIAL ALGEBRA 7 (3–4): 227–82. https://doi.org/10.4171/JCA/75.
Chicago author-date (all authors)
De Schepper, Anneleen, Jeroen Schillewaert, and Hendrik Van Maldeghem. 2023. “A Uniform Characterisation of the Varieties of the Second Row of the Freudenthal-Tits Magic Square over Arbitrary Fields.” JOURNAL OF COMBINATORIAL ALGEBRA 7 (3–4): 227–282. doi:10.4171/JCA/75.
Vancouver
1.
De Schepper A, Schillewaert J, Van Maldeghem H. A uniform characterisation of the varieties of the second row of the Freudenthal-Tits magic square over arbitrary fields. JOURNAL OF COMBINATORIAL ALGEBRA. 2023;7(3–4):227–82.
IEEE
[1]
A. De Schepper, J. Schillewaert, and H. Van Maldeghem, “A uniform characterisation of the varieties of the second row of the Freudenthal-Tits magic square over arbitrary fields,” JOURNAL OF COMBINATORIAL ALGEBRA, vol. 7, no. 3–4, pp. 227–282, 2023.
@article{01HW5NM4RAKQHVE35SZ47HJ6RD,
  abstract     = {{We characterise the projective varieties related to the second row of the Freudenthal- Tits magic square, for both the split and the non-split form, using a common, simple and short geometric axiom system. A special case of our result simultaneously captures the analogues over arbitrary fields of the Severi varieties (comprising the 27-dimensional E6 module and some of its subvarieties), as well as the Veronese representations of projective planes over composition division algebras (most notably the Cayley plane). It is the culmination of almost four decades of work since the original 1984 result by Mazzocca and Melone who characterised the quadric Veronese variety over a finite field of odd order. The latter result is a finite counterpart to the characterisation of the complex quadric Veronese surface by Severi from 1901.}},
  author       = {{De Schepper, Anneleen and Schillewaert, Jeroen and Van Maldeghem, Hendrik}},
  issn         = {{2415-6302}},
  journal      = {{JOURNAL OF COMBINATORIAL ALGEBRA}},
  keywords     = {{Freudenthal-Tits magic square,Cartan variety,Veronese representation,COMBINATORIAL}},
  language     = {{eng}},
  number       = {{3-4}},
  pages        = {{227--282}},
  title        = {{A uniform characterisation of the varieties of the second row of the Freudenthal-Tits magic square over arbitrary fields}},
  url          = {{http://doi.org/10.4171/JCA/75}},
  volume       = {{7}},
  year         = {{2023}},
}
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