no. 4
Symmetric powers of Severi–Brauer varieties
Kollár, János1
1 Princeton University, Princeton NJ 08544-1000, USA
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 4, pp. 849-862.

Nous classons les produits de puissances symétriques d’une variété de Severi–Brauer, à équivalence birationnelle stable près. Notre classification concerne aussi les grassmanniennes, les variétés de drapeaux et les espaces de modules d’applications stables de genre 0.

We classify products of symmetric powers of a Severi–Brauer variety, up to stable birational equivalence. The description also includes Grassmannians, flag varieties and moduli spaces of genus 0 stable maps.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/afst.1584
Kollár, János 1

1 Princeton University, Princeton NJ 08544-1000, USA 
@article{AFST_2018_6_27_4_849_0,
     author = {Koll\'ar, J\'anos},
     title = {Symmetric powers of {Severi{\textendash}Brauer} varieties},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {849--862},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 27},
     number = {4},
     year = {2018},
     doi = {10.5802/afst.1584},
     mrnumber = {3884611},
     zbl = {1423.14092},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/afst.1584/}
}
TY  - JOUR
AU  - Kollár, János
TI  - Symmetric powers of Severi–Brauer varieties
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2018
SP  - 849
EP  - 862
VL  - 27
IS  - 4
PB  - Université Paul Sabatier, Toulouse
UR  - http://www.numdam.org/articles/10.5802/afst.1584/
DO  - 10.5802/afst.1584
LA  - en
ID  - AFST_2018_6_27_4_849_0
ER  - 
%0 Journal Article
%A Kollár, János
%T Symmetric powers of Severi–Brauer varieties
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2018
%P 849-862
%V 27
%N 4
%I Université Paul Sabatier, Toulouse
%U http://www.numdam.org/articles/10.5802/afst.1584/
%R 10.5802/afst.1584
%G en
%F AFST_2018_6_27_4_849_0
Kollár, János. Symmetric powers of Severi–Brauer varieties. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 4, pp. 849-862. doi : 10.5802/afst.1584. http://www.numdam.org/articles/10.5802/afst.1584/

[1] Alexeev, Valery Moduli spaces M g,n (W) for surfaces, Higher dimensional complex varieties (Trento, 1994), Walter de Gruyter (1996), pp. 1-22 | Zbl

[2] Araujo, Carolina; Kollár, János Rational curves on varieties, Higher dimensional varieties and rational points (Budapest, 2001) (Bolyai Society Mathematical Studies), Volume 12, Springer, 2003, pp. 13-68 | DOI | MR | Zbl

[3] Brion, Michel Lectures on the geometry of flag varieties, Topics in cohomological studies of algebraic varieties (Trends in Mathematics), Birkhäuser, 2005, pp. 33-85 | DOI

[4] Dolgachev, Igor V. Rationality of fields of invariants, Algebraic geometry (Bowdoin, 1985) (Proceedings of Symposia in Pure Mathematics), Volume 46, American Mathematical Society (1985), pp. 3-16 | Zbl

[5] Formanek, Edward The ring of generic matrices, J. Algebra, Volume 258 (2002) no. 1, pp. 310-320 | DOI | MR | Zbl

[6] Fulton, William; Pandharipande, Rahul Notes on stable maps and quantum cohomology, Algebraic geometry (Santa Cruz 1995) (Proceedings of Symposia in Pure Mathematics), Volume 62, American Mathematical Society (1995), pp. 45-96 | Zbl

[7] Gille, Philippe; Szamuely, Tamàs Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics, 101, Cambridge University Press, 2006, xi+343 pages | MR | Zbl

[8] Hirschowitz, André La rationalité des schémas de Hilbert de courbes gauches rationnelles suivant Katsylo, Algebraic curves and projective geometry (Trento, 1988) (Lecture Notes in Mathematics), Volume 1389, Springer (1988), pp. 87-90 | DOI | Zbl

[9] Hodge, William Vallance Douglas; Pedoe, Daniel Methods of Algebraic Geometry. Vols. I–III., Cambridge University Press, 1947, 1952, 1954

[10] Hogadi, Amit Products of Brauer-Severi surfaces, Proc. Am. Math. Soc., Volume 137 (2009) no. 1, pp. 45-50 | DOI | MR | Zbl

[11] Katsylo, Pavel I. Rationality of fields of invariants of reducible representations of the group SL 2 , Vestn. Mosk. Univ., Volume 1984 (1984) no. 5, pp. 77-79 | MR | Zbl

[12] Kim, Bumsig; Pandharipande, Rahul The connectedness of the moduli space of maps to homogeneous spaces, Symplectic geometry and mirror symmetry (Seoul, 2000), World Scientific, 2000, pp. 187-201 | Zbl

[13] Kollár, János Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 32, Springer, 1996, viii+320 pages | MR | Zbl

[14] Kollár, János Conics in the Grothendieck ring, Adv. Math., Volume 198 (2005) no. 1, pp. 27-35 | DOI | MR | Zbl

[15] Kollár, János Severi–Brauer varieties; a geometric treatment (2016) (https://arxiv.org/abs/1606.04368)

[16] Kollár, János; Smith, Karen E.; Corti, Alessio Rational and nearly rational varieties, Cambridge Studies in Advanced Mathematics, 92, Cambridge University Press, 2004, vi+235 pages | MR | Zbl

[17] Krashen, Daniel Zero cycles on homogeneous varieties, Adv. Math., Volume 223 (2010) no. 6, pp. 2022-2048 | DOI | MR | Zbl

[18] Krashen, Daniel; Saltman, David J. Severi–Brauer varieties and symmetric powers, Algebraic transformation groups and algebraic varieties (Encyclopaedia of Mathematical Sciences), Volume 132, Springer, 2090, pp. 59-70 | DOI | Zbl

[19] Kresch, Andrew Flag varieties and Schubert calculus, Algebraic groups (Göttingen, 2007) (Mathematisches Institut. Seminare), Universitätsverlag Göttingen, 2007, pp. 73-86 | Zbl

[20] Mattuck, Arthur The field of multisymmetric functions, Proc. Am. Math. Soc., Volume 19 (1968), pp. 764-765 | MR | Zbl

[21] Weil, André Foundations of algebraic geometry, American Mathematical Society Colloquium Publications, XXIX, American Mathematical Society, 1962, xx+363 pages | Zbl

Cité par Sources :