Geometry of the genus 9 Fano 4-folds
[Géométrie des variété de Fano de dimension 4 et genre 9]
Annales de l'Institut Fourier, Tome 60 (2010) no. 4, pp. 1401-1434.

On étudie la géométrie d’une variété générale de Fano de dimension quatre, de genre neuf, et de nombre de Picard un. On calcule son anneau de Chow, et l’on donne une description simple et explicite de sa variété des droites. On utilise alors ces résultats pour étudier des propriétés géométriques de variétés de dimension 3 non quadratiquement normales dans un espace projectif de dimension cinq.

We study the geometry of a general Fano variety of dimension four, genus nine, and Picard number one. We compute its Chow ring and give an explicit description of its variety of lines. We apply these results to study the geometry of non quadratically normal varieties of dimension three in a five dimensional projective space.

DOI : 10.5802/aif.2559
Classification : 14J45, 14J35, 14J60, 14J30, 14M15, 14M07
Keywords: Fano manifold, variety of lines, secant variety, quadratic normality, vector bundles, virtual section, symplectic grassmannian
Mot clés : variété de Fano, variété des droites, variété de secantes, normalité quadratique, fibré vectoriel, section virtuelle, Grassmannienne symplectique
Han, Frédéric 1

1 Université Paris 7 Institut de Mathématiques de Jussieu Case Postale 7012 Bâtiment Chevaleret 75205 Paris Cedex 13 (France)
@article{AIF_2010__60_4_1401_0,
     author = {Han, Fr\'ed\'eric},
     title = {Geometry of the genus 9 {Fano} 4-folds},
     journal = {Annales de l'Institut Fourier},
     pages = {1401--1434},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {60},
     number = {4},
     year = {2010},
     doi = {10.5802/aif.2559},
     zbl = {1203.14043},
     mrnumber = {2722246},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2559/}
}
TY  - JOUR
AU  - Han, Frédéric
TI  - Geometry of the genus 9 Fano 4-folds
JO  - Annales de l'Institut Fourier
PY  - 2010
SP  - 1401
EP  - 1434
VL  - 60
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2559/
DO  - 10.5802/aif.2559
LA  - en
ID  - AIF_2010__60_4_1401_0
ER  - 
%0 Journal Article
%A Han, Frédéric
%T Geometry of the genus 9 Fano 4-folds
%J Annales de l'Institut Fourier
%D 2010
%P 1401-1434
%V 60
%N 4
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2559/
%R 10.5802/aif.2559
%G en
%F AIF_2010__60_4_1401_0
Han, Frédéric. Geometry of the genus 9 Fano 4-folds. Annales de l'Institut Fourier, Tome 60 (2010) no. 4, pp. 1401-1434. doi : 10.5802/aif.2559. http://www.numdam.org/articles/10.5802/aif.2559/

[1] Andreatta, M.; Wisniewski, J. On manifold whose tangent bundle contains a ample subbundle, Invent. math., Volume 146 (2001), pp. 209-217 | DOI | MR | Zbl

[2] Barth, W. Irreducibility of the space of mathematical instanton bundles with rank 2, c 2 =4, Math Ann., Volume 258 (1981), pp. 81-106 | DOI | MR | Zbl

[3] Gruson, L.; Peskine, C. Courbes de l’espace projectif, variétés de sécantes. Enumerative geometry and classical algebraic geometry, Nice (1981) Prog Math, 24, Birkhäuser. Boston, 1982 | Zbl

[4] Hartshorne, R. Algebraic Geometry, 52, Springer-Verlag GTM, 1977 | MR | Zbl

[5] Hartshorne, R. Stable reflexive sheaves, Math Ann, Volume 254 (1980), pp. 121-176 | DOI | MR | Zbl

[6] Iliev, A. The SP 3 -Grassmannian and duality for prime Fano threefolds of genus 9, Manuscripta math., Volume 112 (2003), pp. 29-53 | DOI | MR | Zbl

[7] Iliev, A.; Manivel, L. Severi varieties and their varieties of reduction, J. reine angew. Math, Volume 585 (2005), pp. 93-139 | DOI | MR | Zbl

[8] Iliev, A.; Ranestad, K. Geometry of the Lagrangian Grassmannian LG(3,6) with applications to Brill-Noether loci, Mich. Math. Journal, Volume 53 (2005), pp. 383-417 | DOI | MR | Zbl

[9] Kac, V. G. Some remarks on nilpotent orbits, Journal of Algebra, Volume 64 (1980), pp. 190-213 Math. 32 (1995) | DOI | MR | Zbl

[10] Kollar, J. Rationnal curves on algebraic varieties., Ergebnisse der Math., 32, Springer-Verlag, 1995 | MR | Zbl

[11] Kuznetsov, A. Hyperplane sections and derived categories, Izvestiya Mathematics, Volume 70 (2006) no. 3, p. 447-447 | DOI | MR | Zbl

[12] Manivel, L. Configuration of lines and models of Lie algebras, Journal of Algebra, Volume 304 (2006), pp. 457-486 | DOI | MR | Zbl

[13] Manivel, L.; Mezzetti, E. On linear spaces of skew-symmetric matrices of constant rank, Manuscripta math., Volume 117 (2005), pp. 319-331 | DOI | MR | Zbl

[14] Mezzetti, E.; de Poi, P. Congruences of lines in IP 5 , quadratic normality, and completely exceptional Monge-Ampère equations, Geom Dedicata, Volume 131 (2008), pp. 213-230 | DOI | MR | Zbl

[15] Okonek, C.; Schneider, M.; Spindler, H. Vector bundles on complex projective spaces, Progress in Math 3, Birkäuser Boston Mass, 1980 | MR | Zbl

[16] Đoković, D. Z.; Osterloh, A. On polynomial invariants of several qubits, J. Math. Phys., Volume 50 (2009) no. 3, 033509, pp. 81-106 | MR

[17] Tjurin, A. N. On the superpositions of mathematical instantons, In Artin, Tate, J.(eds) Arithmetic and geometry. Prog. Math, Volume 36 (1983), pp. 433-450 (Birkhäuser) | MR | Zbl

[18] Weyman, J. Cohomology of vector bundles and syzygies, Tracts in Mathematics, 149, Cambridge, 2003 | MR | Zbl

Cité par Sources :