On Severi varieties as intersections of a minimum number of quadrics

  • Hendrik Van Maldeghem Ghent University, Department of Mathematics: Algebra & Geometry, Krijgslaan 281, S25, B-9000 Gent, Belgium.
  • Magali Victoor Ghent University, Department of Mathematics: Algebra & Geometry, Krijgslaan 281, S25, B-9000 Gent, Belgium.
Keywords: Cartan variety, quadrics, exceptional geometry, Severi variety, quaternion veronesian

Abstract

Let \({\mathscr{V}}\) be a variety related to the second row of the Freudenthal-Tits Magic square in \(N\)-dimensional projective space over an arbitrary field. We show that there exist \(M\leq N\) quadrics intersecting precisely in \({\mathscr{V}}\) if and only if there exists a subspace of projective dimension \(N-M\) in the secant variety disjoint from the Severi variety. We present some examples of such subspaces of relatively large dimension. In particular, over the real numbers we show that the Cartan variety (related to the exceptional group \({E_6}\)\((\mathbb R)\)) is the set-theoretic intersection of 15 quadrics.

References

M. Aschbacher, “The 27-dimensional module for E6. I.”, Invent. Math., vol. 89, no. 1, pp. 159–195, 1987.

S. G. Barwick, W.-A. Jackson and P. Wild, “The Bose representation of PG(2, q3) in PG(8, q)”, Australas. J. Combin., vol. 79, pp. 31–54, 2021.

M. Brion, “Représentations exceptionelles des groupes semi-simple”, Ann. Sci. École Norm. Sup. (4), vol. 18, no. 2, pp. 345–387, 1985.

A. M. Cohen, Diagram Geometry, related to Lie algebras and groups, book in preparation, see http://arpeg.nl/wp-content/uploads/2020/09/book2n.pdf.

D. Eisenbud and E. G. Evans, “Every algebraic set in n-space is the intersection of n hyper-surfaces”, Invent. Math., vol. 19, pp. 107–112, 1973.

J. W. P. Hirschfeld and J. A. Thas, General Galois geometries, Springer Monographs in Mathematics, London: Springer-Verlag, 2016.

W. Lichtenstein, “A system of quadrics describing the orbit of the highest weight vector”, Proc. Amer. Math. Soc., vol. 84, no. 4, pp. 605–608, 1982.

S. E. Payne and J. A. Thas, Finite generalized quadrangles, EMS Series of Lectures in Mathematics, Zu ̈rich: European Mathematical Society (EMS), 2009.

J. Schillewaert and H. Van Maldeghem, “On the varieties of the second row of the split Freudenthal-Tits magic square”, Ann. Inst. Fourier (Grenoble), vol. 67, no. 6, pp. 2265–2305, 2017.

A. Ramanathan, “Equations defining Schubert varieties and Frobenius splitting of diagonals”, Inst. Hautes Études Sci. Publ. Math., no. 65, pp. 61–90, 1987.

H. Van Maldeghem and M. Victoor, “Some combinatorial and geometric constructions of spherical buildings” in Surveys in combinatorics 2019, London Math. Soc. Lecture Note Ser., 456, Cambridge: Cambridge Univ. Press, pp. 237–265, 2019.

Published
2022-08-22
How to Cite
[1]
H. Van Maldeghem and M. Victoor, “On Severi varieties as intersections of a minimum number of quadrics”, CUBO, vol. 24, no. 2, pp. 307–331, Aug. 2022.
Section
Articles