# Geometry of the genus 9 Fano 4-folds

Frédéric Han^{[1]}

- [1] Université Paris 7 Institut de Mathématiques de Jussieu Case Postale 7012 Bâtiment Chevaleret 75205 Paris Cedex 13 (France)

Annales de l’institut Fourier (2010)

- Volume: 60, Issue: 4, page 1401-1434
- ISSN: 0373-0956

## Access Full Article

top## Abstract

top## How to cite

topHan, Frédéric. "Geometry of the genus 9 Fano 4-folds." Annales de l’institut Fourier 60.4 (2010): 1401-1434. <http://eudml.org/doc/116308>.

@article{Han2010,

abstract = {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.},

affiliation = {Université Paris 7 Institut de Mathématiques de Jussieu Case Postale 7012 Bâtiment Chevaleret 75205 Paris Cedex 13 (France)},

author = {Han, Frédéric},

journal = {Annales de l’institut Fourier},

keywords = {Fano manifold; variety of lines; secant variety; quadratic normality; vector bundles; virtual section; symplectic grassmannian; vector bundle; family of lines; symplectic Grassmannian},

language = {eng},

number = {4},

pages = {1401-1434},

publisher = {Association des Annales de l’institut Fourier},

title = {Geometry of the genus 9 Fano 4-folds},

url = {http://eudml.org/doc/116308},

volume = {60},

year = {2010},

}

TY - JOUR

AU - Han, Frédéric

TI - Geometry of the genus 9 Fano 4-folds

JO - Annales de l’institut Fourier

PY - 2010

PB - Association des Annales de l’institut Fourier

VL - 60

IS - 4

SP - 1401

EP - 1434

AB - 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.

LA - eng

KW - Fano manifold; variety of lines; secant variety; quadratic normality; vector bundles; virtual section; symplectic grassmannian; vector bundle; family of lines; symplectic Grassmannian

UR - http://eudml.org/doc/116308

ER -

## References

top- M. Andreatta, J. Wisniewski, On manifold whose tangent bundle contains a ample subbundle, Invent. math. 146 (2001), 209-217 Zbl1081.14060MR1859022
- W. Barth, Irreducibility of the space of mathematical instanton bundles with rank 2, ${c}_{2}=4$, Math Ann. 258 (1981), 81-106 Zbl0477.14014MR641670
- L. Gruson, C. Peskine, Courbes de l’espace projectif, variétés de sécantes. Enumerative geometry and classical algebraic geometry, 24 (1982), Birkhäuser. Boston Zbl0531.14020
- R. Hartshorne, Algebraic Geometry, 52 (1977), Springer-Verlag GTM Zbl0367.14001MR463157
- R. Hartshorne, Stable reflexive sheaves, Math Ann 254 (1980), 121-176 Zbl0431.14004MR597077
- A. Iliev, The $S{P}_{3}$-Grassmannian and duality for prime Fano threefolds of genus $9$, Manuscripta math. 112 (2003), 29-53 Zbl1078.14528MR2005929
- A. Iliev, L. Manivel, Severi varieties and their varieties of reduction, J. reine angew. Math 585 (2005), 93-139 Zbl1083.14060MR2164624
- A. Iliev, K. Ranestad, Geometry of the Lagrangian Grassmannian LG(3,6) with applications to Brill-Noether loci, Mich. Math. Journal 53 (2005), 383-417 Zbl1084.14042MR2152707
- V. G. Kac, Some remarks on nilpotent orbits, Journal of Algebra 64 (1980), 190-213 Zbl0431.17007MR575790
- J. Kollar, Rationnal curves on algebraic varieties., 32 (1995), Springer-Verlag Zbl0877.14012MR1440180
- A. Kuznetsov, Hyperplane sections and derived categories, Izvestiya Mathematics 70 (2006), 447-447 Zbl1133.14016MR2238172
- L. Manivel, Configuration of lines and models of Lie algebras, Journal of Algebra 304 (2006), 457-486 Zbl1167.17001MR2256401
- L. Manivel, E. Mezzetti, On linear spaces of skew-symmetric matrices of constant rank, Manuscripta math. 117 (2005), 319-331 Zbl1084.14050MR2154253
- E. Mezzetti, P. de Poi, Congruences of lines in $\mathrm{I}\phantom{\rule{-0.166667em}{0ex}}{\mathrm{P}}_{5}$, quadratic normality, and completely exceptional Monge-Ampère equations, Geom Dedicata 131 (2008), 213-230 Zbl1185.14042MR2369200
- C. Okonek, M. Schneider, H. Spindler, Vector bundles on complex projective spaces, (1980), Birkäuser Boston Mass Zbl0438.32016MR561910
- D. Z. Đoković, A. Osterloh, On polynomial invariants of several qubits, J. Math. Phys. 50 (2009), 81-106 Zbl1202.81017MR2510914
- A. N. Tjurin, On the superpositions of mathematical instantons, In Artin, Tate, J.(eds) Arithmetic and geometry. Prog. Math 36 (1983), 433-450 Zbl0541.14013MR717619
- J. Weyman, Cohomology of vector bundles and syzygies, 149 (2003), Cambridge Zbl1075.13007MR1988690

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.