logo AFST
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.

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
János Kollár 1

1 Princeton University, Princeton NJ 08544-1000, USA
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
@article{AFST_2018_6_27_4_849_0,
     author = {J\'anos Koll\'ar},
     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},
     zbl = {1423.14092},
     mrnumber = {3884611},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1584/}
}
TY  - JOUR
AU  - János Kollár
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  - https://afst.centre-mersenne.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 János Kollár
%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 https://afst.centre-mersenne.org/articles/10.5802/afst.1584/
%R 10.5802/afst.1584
%G en
%F AFST_2018_6_27_4_849_0
János Kollár. 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. https://afst.centre-mersenne.org/articles/10.5802/afst.1584/

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

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

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

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

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

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

[8] André Hirschowitz 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 (1988), pp. 87-90 | DOI | Zbl

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

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

[11] Pavel I. Katsylo 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] Bumsig Kim; Rahul Pandharipande 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] János Kollár Rational curves on algebraic varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 32, Springer, 1996, viii+320 pages | MR | Zbl

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

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

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

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

[18] Daniel Krashen; David J. Saltman 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] Andrew Kresch Flag varieties and Schubert calculus, Algebraic groups (Göttingen, 2007) (Mathematisches Institut. Seminare), Universitätsverlag Göttingen, 2007, pp. 73-86 | Zbl

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

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

Cité par Sources :