no. 6
On the varieties of the second row of the split Freudenthal–Tits Magic Square
[Sur les variétés de la deuxième rangée du carré magique deployé de Freudenthal–Tits]
Schillewaert, Jeroen1 ; Van Maldeghem, Hendrik2
1 Department of Mathematics, University of Auckland, Private Bag 92019 Auckland (New Zealand)
2 Department of Mathematics, Ghent University, Krijgslaan 281-S25, 9000 Ghent (Belgium)
Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2265-2305.

Le but principal de cet article est de fournir une caractérisation géométrique des analogues sur les corps quelconques des quatre variétés complexes de Severi, c’est-à-dire la surface de Veronese, la variété de Segre S(2,2), la grassmannienne G(2,6) et la variété exceptionnelle de type E 6 . Notre théorème peut être vu comme une généralisation considérable de l’approche de Mazzocca et Melone pour les surfaces de Veronese sur les corps finis. Cette approche utilise des analogues combinatoires de certaines propriétés, qui expriment que les variétés de Severi complexes sont lisses, comme axiomes.

Our main aim is to provide a uniform geometric characterization of the analogues over arbitrary fields of the four complex Severi varieties, i.e. the quadric Veronese varieties in 5-dimensional projective spaces, the Segre varieties in 8-dimensional projective spaces, the line Grassmannians in 14-dimensional projective spaces, and the exceptional varieties of type E 6 in 26-dimensional projective space. Our theorem can be regarded as a far-reaching generalization of Mazzocca and Melone’s approach to finite quadric Veronesean varieties. This approach uses combinatorial analogues of smoothness properties of complex Severi varieties as axioms.

Reçu le :
Révisé le :
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DOI : 10.5802/aif.3136
Classification : 51E24, 51A45, 14M12, 17C37, 20G15
Keywords: Severi variety, Veronese variety, Segre variety, Grassmann variety, Tits-building
Mot clés : Variété de Severi, variété de Veronese, variété de Segre, grassmanniene, immeuble de Tits

Schillewaert, Jeroen 1 ; Van Maldeghem, Hendrik 2

1 Department of Mathematics, University of Auckland, Private Bag 92019 Auckland (New Zealand)
2 Department of Mathematics, Ghent University, Krijgslaan 281-S25, 9000 Ghent (Belgium)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Schillewaert, Jeroen; Van Maldeghem, Hendrik. On the varieties of the second row of the split Freudenthal–Tits Magic Square. Annales de l'Institut Fourier, Tome 67 (2017) no. 6, pp. 2265-2305. doi : 10.5802/aif.3136. https://aif.centre-mersenne.org/articles/10.5802/aif.3136/

[1] Aschbacher, Michael The 27-dimensional module for E 6 , I., Invent. Math., Volume 89 (1987), pp. 159-195 | DOI | Zbl

[2] Bourbaki, Nicolas Groupes et Algèbres de Lie, Chapitres IV, V and VI, Actualités Scientifiques et Industrielles, 1337, Hermann, 1968 | Zbl

[3] Brouwer, Andries E.; Cohen, Arjeh M. Some remarks on Tits geometries., Indag. Math., Volume 45 (1983), pp. 393-402 | DOI | Zbl

[4] Buekenhout, Francis; Cameron, Peter Projective and Affine Geometry over Division Rings, Handbook of Incidence Geometry, North-Holland, 1995, pp. 27-62 | Zbl

[5] Chaput, Pierre-Emmanuel Severi varieties, Math. Z., Volume 240 (2002) no. 2, pp. 451-459 | DOI | Zbl

[6] Chevalley, Claude The Algebraic Theory of Spinors, Columbia University Press, 1954, viii+131 pages | Zbl

[7] Cooperstein, Bruce N.; Shult, Ernest E. Frames and bases of Lie incidence geometries, J. Geom., Volume 60 (1997) no. 1-2, pp. 17-46 | DOI | Zbl

[8] De Bruyn, Bart; Van Maldeghem, Hendrik Dual polar spaces of rank 3 defined over quadratic alternative division algebras, J. Reine Angew. Math., Volume 715 (2016), pp. 39-74 | Zbl

[9] Havlicek, Hans Zur Theorie linearer Abbildungen I, II., J. Geom., Volume 16 (1981), pp. 152-180 | DOI | Zbl

[10] Hirschfeld, James William Peter; Thas, Joseph Adolphe General Galois Geometries, Oxford Mathematical Monographs, Clarendon Press, 1991, xii+407 pages | Zbl

[11] Iliev, Atanas; Markushevich, Dimitri Elliptic curves and rank-2 vector bundles on the prime Fano threefold of genus 7, Adv. Geom., Volume 4 (2004) no. 3, pp. 287-318 | DOI | Zbl

[12] Ionescu, Paltin; Russo, Francesco Varieties with quadratic entry locus II., Compos. Math., Volume 144 (2008) no. 4, pp. 949-962 | DOI | Zbl

[13] Ionescu, Paltin; Russo, Francesco Conic-connected manifolds, J. Reine Angew. Math., Volume 644 (2010), pp. 145-158 | Zbl

[14] Kasikova, Anna; Shult, Ernest E. Absolute embeddings of point-line geometries, J. Algebra, Volume 238 (2001) no. 1, pp. 265-291 | DOI | Zbl

[15] Krauss, O.; Schillewaert, Jeroen; Van Maldeghem, Hendrik Veronesean representations of Moufang planes, Mich. Math. J., Volume 64 (2015), pp. 819-847 | DOI | Zbl

[16] Lazarsfeld, Robert; Van de Ven, Antonius Topics in the Geometry of Projective Space. Recent work of F. L. Zak, DMV Seminar, 4, Birkhäuser, 1984 | Zbl

[17] Manivel, Laurent On spinor varieties and their secants, SIGMA Symmetry Integrability Geom. Methods Appl., Volume 5 (2009) (Paper 078, 22 pp.) | Zbl

[18] Mazzocca, Francesco; Melone, Nicola Caps and Veronese varieties in projective Galois spaces, Discrete Math., Volume 48 (1984), pp. 243-252 | DOI | Zbl

[19] Mukai, Shigeru Curves and symmetric spaces, I., Am. J. Math., Volume 117 (1995) no. 6, pp. 1627-1644 | DOI | Zbl

[20] Nash, Oliver K-theory, LQEL manifolds and Severi varieties, Geom. Topol., Volume 18 (2014) no. 3, pp. 1245-1260 | DOI | Zbl

[21] Russo, Francesco Varieties with quadratic entry locus I., Math. Ann., Volume 344 (2009) no. 3, pp. 597-617 | DOI | Zbl

[22] Schillewaert, Jeroen; Van Maldeghem, Hendrik Hermitian Veronesean caps, Buildings, finite geometries and groups (ICM, Hyderabad, India, 2010) (Springer Proceedings in Mathematics), Volume 10 (2012), pp. 175-191 | Zbl

[23] Schillewaert, Jeroen; Van Maldeghem, Hendrik Quadric Veronesean caps, Bull. Belg. Math. Soc. Simon Stevin, Volume 20 (2013) no. 1, pp. 19-25 | Zbl

[24] Schillewaert, Jeroen; Van Maldeghem, Hendrik Imbrex geometries, J. Combin. Theory Ser. A, Volume 127 (2014), pp. 286-302 | DOI | Zbl

[25] Schillewaert, Jeroen; Van Maldeghem, Hendrik Projective planes over quadratic 2-dimensional algebras, Adv. Math., Volume 262 (2014), pp. 784-822 | DOI | Zbl

[26] Schillewaert, Jeroen; Van Maldeghem, Hendrik A combinatorial characterization of the Lagrangian Grassmannian LG(3,6), Glasg. Math. J., Volume 58 (2016) no. 2, pp. 293-311 | DOI | Zbl

[27] Shult, Ernest E. Points and Lines. Characterizing the Classical Geometries, Universitext, Springer, 2011, xxii+676 pages | Zbl

[28] Thas, Joseph Adolphe; Van Maldeghem, Hendrik Characterizations of the finite quadric Veroneseans 𝒱 n 2 n , Q. J. Math., Volume 55 (2004), pp. 99-113 | DOI | Zbl

[29] Tits, Jacques Sur certaines classes d’espaces homogènes de groupes de Lie, Acad. Roy. Belg. Cl. Sci. Mém. Collect., 8 o, 29(3), Palais des académies, 1955, 268 pages | Zbl

[30] Tits, Jacques Sur la géometrie des R-espaces, J. Math. Pures Appl., Volume 36 (1957), pp. 17-38 | Zbl

[31] Tits, Jacques Algèbres alternatives, algèbres de Jordan et algèbres de Lie exceptionnelles, Nederl. Akad. Wet., Proc., Volume 28 (1966), pp. 223-237 | DOI | Zbl

[32] Tits, Jacques Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Mathematics, 386, Springer, 1974, x+299 pages | Zbl

[33] Tits, Jacques A local approach to buildings, The Geometric Vein. The Coxeter Festschrift (1982), pp. 519-547 | Zbl

[34] Wells, Albert L.jun. Universal projective embeddings of the Grassmannian, half spinor, and dual orthogonal geometries, Q. J. Math., Oxf., Volume 34 (1983), pp. 375-386 | DOI | Zbl

[35] Zak, Fyodor L. Tangents and Secants of Algebraic Varieties, Translation of Mathematical Monographs, 127, American Mathematical Society, 1983, vii+164 pages | Zbl

[36] Zanella, Corrado Universal properties of the Corrado-Segre embedding, Bull. Belg. Math. Soc. Simon Stevin, Volume 3 (1996) no. 1, pp. 65-79 | Zbl

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