Fano congruences of index 3 and alternating 3-forms
Annales de l'Institut Fourier, Volume 67 (2017) no. 5, p. 2099-2165

We study congruences of lines X ω defined by a sufficiently general choice of an alternating 3-form ω in n+1 dimensions, as Fano manifolds of index 3 and dimension n-1. These congruences include the G 2 -variety for n=6 and the variety of reductions of projected 2 × 2 for n=7.

We compute the degree of X ω as the n-th Fine number and study the Hilbert scheme of these congruences proving that the choice of ω bijectively corresponds to X ω except when n=5. The fundamental locus of the congruence is also studied together with its singular locus: these varieties include the Coble cubic for n=8 and the Peskine variety for n=9.

The residual congruence Y of X ω with respect to a general linear congruence containing X ω is analysed in terms of the quadrics containing the linear span of X ω . We prove that Y is Cohen–Macaulay but non-Gorenstein in codimension 4. We also examine the fundamental locus G of Y of which we determine the singularities and the irreducible components.

Nous étudions des congruences de droites X ω définies par une 3-forme alternée suffisamment générale en n+1 variables. Celles-ci sont des variétés de Fano d’indice 3 et dimension n-1. La classe de ces congruences contient la 5-variété homogène sous G 2 dans 13 pour n=6 et la variété des réductions d’une projection générique de 2 × 2 dans 7 pour n=7.

Nous montrons que le degré de X ω est le n-ième nombre de Fine. Nous étudions le schéma de Hilbert de ces congruences et montrons que le choix de ω correspond birationnellement au choix de X ω sauf si n=5.

Le lieu fondamental de ces congruences est étudié aussi bien que son lieu singulier  : la classe de ces variétés inclut la cubique de Coble pour n=8 et la variété de Peskine pour n=9.

La congruence résiduelle Y de X ω par rapport à une congruence linéaire générique contenant X ω est analysée à travers les quadriques qui contiennent l’espace linéaire engendré par X ω . Nous montrons que Y est Cohen–Macaulay mais pas Gorenstein en codimension 4. Nous examinons le lieu fondamental G de Y, duquel nous déterminons les singularités et les composantes irréductibles.

Received : 2016-06-24
Revised : 2016-12-19
Accepted : 2017-01-24
Published online : 2017-11-17
DOI : https://doi.org/10.5802/aif.3131
Classification:  14M15,  14J45,  14J60,  14M06,  14M05
Keywords: Fano varieties; congruences of lines; trivectors; alternating 3-forms; Cohen–Macaulay varieties; linkage; linear congruences; Coble variety; Peskine variety; variety of reductions; secant lines; fundamental loci.
@article{AIF_2017__67_5_2099_0,
     author = {De Poi, Pietro and Faenzi, Daniele and Mezzetti, Emilia and Ranestad, Kristian},
     title = {Fano congruences of index 3 and alternating 3-forms},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {5},
     year = {2017},
     pages = {2099-2165},
     doi = {10.5802/aif.3131},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2017__67_5_2099_0}
}
De Poi, Pietro; Faenzi, Daniele; Mezzetti, Emilia; Ranestad, Kristian. Fano congruences of index 3 and alternating 3-forms. Annales de l'Institut Fourier, Volume 67 (2017) no. 5, pp. 2099-2165. doi : 10.5802/aif.3131. http://www.numdam.org/item/AIF_2017__67_5_2099_0/

[1] Sloane, Neil James A. The Online Encyclopedia of Integer Sequences (https://oeis.org/A000108 )

[2] Abo, Hirotachi; Ottaviani, Giorgio; Peterson, Chris Non-defectivity of Grassmannians of planes, J. Algebr. Geom., Tome 21 (2012) no. 1, pp. 1-20 | Article | MR 2846677 | Zbl 1242.14050

[3] Agafonov, Sergey I.; Ferapontov, Evgeny V. Systems of conservation laws of Temple class, equations of associativity and linear congruences in 4 , Manuscr. Math., Tome 106 (2001) no. 4, pp. 461-488 | Article | MR 1875343 | Zbl 1149.35385

[4] Bastianelli, Francesco; Cortini, Renza; De Poi, Pietro The gonality theorem of Noether for hypersurfaces, J. Algebr. Geom., Tome 23 (2014) no. 2, pp. 313-339 | Article | MR 3166393 | Zbl 1317.14029

[5] Bastianelli, Francesco; De Poi, Pietro; Ein, Lawrence; Lazarsfeld, Robert; Ullery, Brooke Measures of irrationality for hypersurfaces of large degree, Compos. Math., Tome 153 (2017), pp. 2368-2393 | Article

[6] Ciliberto, Ciro; Mella, Massimiliano; Russo, Francesco Varieties with one apparent double point, J. Algebr. Geom., Tome 13 (2004) no. 3, pp. 475-512 | Article | MR 2047678 | Zbl 1077.14076

[7] Ciliberto, Ciro; Russo, Francesco On the classification of OADP varieties, Sci. China Math., Tome 54 (2011) no. 8, pp. 1561-1575 | Article | MR 2824959 | Zbl 1246.14068

[8] De Poi, Pietro On first order congruences of lines of 4 with a fundamental curve, Manuscr. Math., Tome 106 (2001) no. 1, pp. 101-116 (erratum ibid 127 (2008), no. 1, p. 137) | Article | MR 1860982 | Zbl 1066.14062

[9] De Poi, Pietro Threefolds in 5 with one apparent quadruple point, Commun. Algebra, Tome 31 (2003) no. 4, pp. 1927-1947 | Article | MR 1972898 | Zbl 1018.14015

[10] De Poi, Pietro; Mezzetti, Emilia Linear congruences and hyperbolic systems of conservation laws, Projective varieties with unexpected properties, Walter de Gruyter (2005), pp. 209-230 | MR 2202254 | Zbl 1101.14062

[11] De Poi, Pietro; Mezzetti, Emilia On congruences of linear spaces of order one, Rend. Ist. Mat. Univ. Trieste, Tome 39 (2007), pp. 177-206 | MR 2441617 | Zbl 1151.14331

[12] De Poi, Pietro; Mezzetti, Emilia Congruences of lines in 5 , quadratic normality, and completely exceptional Monge-Ampère equations, Geom. Dedicata, Tome 131 (2008), pp. 213-230 | Article | MR 2369200 | Zbl 1185.14042

[13] Debarre, Olivier; Voisin, Claire Hyper-Kähler fourfolds and Grassmann geometry, J. Reine Angew. Math., Tome 649 (2010), pp. 63-87 | Article | MR 2746467 | Zbl 1217.14028

[14] Deutsch, Emeric; Shapiro, Louis A survey of the Fine numbers, Discrete Math., Tome 241 (2001) no. 1-3, pp. 241-265 (Selected papers in honor of Helge Tverberg) | Article | MR 1861421 | Zbl 0992.05011

[15] Djoković, Dragomir Ž. Closures of equivalence classes of trivectors of an eight-dimensional complex vector space, Can. Math. Bull., Tome 26 (1983) no. 1, pp. 92-100 | Article | MR 681957 | Zbl 0471.22008

[16] Faenzi, Daniele; Fania, Maria Lucia Skew-symmetric matrices and Palatini scrolls, Math. Ann., Tome 347 (2010) no. 4, pp. 859-883 | Article | MR 2658146 | Zbl 1200.14029

[17] Fulton, William; Harris, Joe Representation theory, Springer, Graduate Texts in Mathematics, Tome 129 (1991), xvi+551 pages (A first course, Readings in Mathematics) | Article | MR 1153249 | Zbl 0744.22001

[18] Gruson, Laurent; Sam, Steven V. Alternating trilinear forms on a nine-dimensional space and degenerations of (3,3)-polarized Abelian surfaces, Proc. Lond. Math. Soc., Tome 110 (2015) no. 3, pp. 755-785 | Article | MR 3342104 | Zbl 1354.14019

[19] Gruson, Laurent; Sam, Steven V.; Weyman, Jerzy Moduli of abelian varieties, Vinberg θ-groups, and free resolutions, Commutative algebra, Springer (2013), pp. 419-469 | Article | MR 3051381 | Zbl 1274.13027

[20] Gurevich, Grigorii B. Classification of tri-vectors of rank 8, Dokl. Akad. Nauk. SSSR, Tome 2 (1935), pp. 353-355 | Zbl 0012.10003

[21] Gurevich, Grigorii B. Foundations of the theory of algebraic invariants, P. Noordhoff Ltd. (1964) | Zbl 0128.24601

[22] Han, Frédéric Duality and quadratic normality, Rend. Ist. Mat. Univ. Trieste, Tome 47 (2015), pp. 9-16 | MR 3456574 | Zbl 1355.14031

[23] Harris, Joe; Tu, Loring W. On symmetric and skew-symmetric determinantal varieties, Topology, Tome 23 (1984) no. 1, pp. 71-84 | Article | MR 721453 | Zbl 0534.55010

[24] Holweck, Frédéric Singularities of duals of Grassmannians, J. Algebra, Tome 337 (2011) no. 1, pp. 369-384 | Article | MR 2796082 | Zbl 1244.14039

[25] Iliev, Atanas; Manivel, Laurent Severi varieties and their varieties of reductions, J. Reine Angew. Math., Tome 585 (2005), pp. 93-139 | Article | MR 2164624 | Zbl 1083.14060

[26] Kapustka, Michał; Ranestad, Kristian Vector bundles on Fano varieties of genus ten, Math. Ann., Tome 356 (2013) no. 2, pp. 439-467 | Article | MR 3048603 | Zbl 1279.14015

[27] Kleiman, Steven L. The transversality of a general translate, Compos. Math., Tome 28 (1974), pp. 287-297 | MR 0360616 | Zbl 0288.14014

[28] Migliore, Juan C. Introduction to liaison theory and deficiency modules, Birkhäuser, Progress in Mathematics, Tome 165 (1998), xii+215 pages | Article | MR 1712469 | Zbl 0921.14033

[29] Mukai, Shigeru Biregular classification of Fano 3-folds and Fano manifolds of coindex 3, Proc. Natl. Acad. Sci. U.S.A., Tome 86 (1989) no. 9, pp. 3000-3002 | Article | MR MR995400 (90g:14024) | Zbl 0679.14020

[30] Mukai, Shigeru Curves and Grassmannians, Algebraic geometry and related topics (Inchon, 1992), International Press (Conference Proceedings and Lecture Notes in Algebraic Geometry) Tome 1 (1993), pp. 19-40 | MR 1285374 | Zbl 0846.14030

[31] Ottaviani, Giorgio On Cayley bundles on the five-dimensional quadric, Boll. Unione Mat. Ital., Tome 4 (1990) no. 1, pp. 87-100 | MR 1047517 | Zbl 0722.14006

[32] Ottaviani, Giorgio On 3-folds in 5 which are scrolls, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Tome 19 (1992) no. 3, pp. 451-471 | MR 1205407 | Zbl 0786.14026

[33] Ozeki, Ikuzō On the microlocal structure of a regular prehomogeneous vector space associated with GL (8), Proc. Japan Acad., Tome 56 (1980) no. 1, pp. 18-21 http://projecteuclid.org/euclid.pja/1195517030 | Article | MR 562863 | Zbl 0454.58014

[34] Peskine, Christian Order 1 congruences of lines with smooth fundamental scheme, Rend. Ist. Mat. Univ. Trieste, Tome 47 (2015), pp. 203-216 | MR 3456584 | Zbl 1350.14040

[35] Schouten, Jan Arnoldus Klassifizierung der alternierenden Grössen dritten Grades in 7 dimensionen, Rendiconti Palermo, Tome 55 (1931), pp. 137-156 | Article | Zbl 57.0975.01

[36] Segre, Corrado Sui complessi lineari di piani nello spazio a cinque dimensioni, Ann. Mat. Pura Appl., Tome 7 (1917), pp. 75-123 | Zbl 46.1023.01

[37] Vinberg, Èrnest B.; Èlašvili, Alexander G. Classification of trivectors of a 9-dimensional space, Sel. Math. Sov., Tome 7 (1978) no. 1, pp. 63-98 | MR 504529 | Zbl 0648.15021

[38] Weyman, Jerzy Cohomology of vector bundles and syzygies, Cambridge University Press, Cambridge Tracts in Mathematics, Tome 149 (2003), xiv+371 pages | Article | MR 1988690 | Zbl 1075.13007