Comptes Rendus
no. 12
Algebraic geometry
Twisted cubic curves in the Segre variety
[Courbes rationnelles dans la variété de Segre]

Présenté par : Claire Voisin

Kiryong Chung1 ; Wanseok Lee2 
1 Department of Mathematics Education, Kyungpook National University, 80 Daehakro, Bukgu, Daegu 41566, Republic of Korea
2 Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of Korea
Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1123-1127.

Soit X=P1×P1×P1 la variété de Segre. Soit S l'espace des courbes cubiques rationnelles de tridegré (1,1,1) dans X. Dans cet article, nous prouvons que S est une variété rationnelle, lisse, de dimension 6. Nous calculons également le polynôme de Poincaré de S à l'aide d'une stratification dont les strates sont des fibrés projectifs.

Let X=P1×P1×P1 be the Segre variety. Let S be the space of twisted cubic curves in X with tri-degree (1,1,1). In this note, we prove that S is a rational, smooth variety of dimension 6. Also, we compute the Poincaré polynomial of S by stratifying the space into projective space fibration over some base spaces.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.09.008
Mots clés : Rational curves, Stable maps, Stable sheaves
Kiryong Chung 1 ; Wanseok Lee 2

1 Department of Mathematics Education, Kyungpook National University, 80 Daehakro, Bukgu, Daegu 41566, Republic of Korea
2 Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of Korea 
@article{CRMATH_2015__353_12_1123_0,
     author = {Kiryong Chung and Wanseok Lee},
     title = {Twisted cubic curves in the {Segre} variety},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1123--1127},
     publisher = {Elsevier},
     volume = {353},
     number = {12},
     year = {2015},
     doi = {10.1016/j.crma.2015.09.008},
     language = {en},
}
TY  - JOUR
AU  - Kiryong Chung
AU  - Wanseok Lee
TI  - Twisted cubic curves in the Segre variety
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 1123
EP  - 1127
VL  - 353
IS  - 12
PB  - Elsevier
DO  - 10.1016/j.crma.2015.09.008
LA  - en
ID  - CRMATH_2015__353_12_1123_0
ER  - 
%0 Journal Article
%A Kiryong Chung
%A Wanseok Lee
%T Twisted cubic curves in the Segre variety
%J Comptes Rendus. Mathématique
%D 2015
%P 1123-1127
%V 353
%N 12
%I Elsevier
%R 10.1016/j.crma.2015.09.008
%G en
%F CRMATH_2015__353_12_1123_0
Kiryong Chung; Wanseok Lee. Twisted cubic curves in the Segre variety. Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1123-1127. doi : 10.1016/j.crma.2015.09.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.09.008/

[1] B. Bakker; A. Jorza Higher rank stable pairs on K3 surfaces, Commun. Number Theory Phys., Volume 6 (2012), pp. 805-847

[2] E. Ballico; S. Huh Curves in Segre threefolds | arXiv

[3] K. Chung; Y.-H. Kiem Hilbert scheme of rational cubic curves via stable maps, Amer. J. Math., Volume 133 (2011) no. 3, pp. 797-834

[4] K. Chung; J. Hong; Y.-H. Kiem Compactified moduli spaces of rational curves in projective homogeneous varieties, J. Math. Soc. Jpn., Volume 64 (2012) no. 4, pp. 1211-1248 (MR 2998922)

[5] D.R. Grayson; M.E. Stillman Macaulay2, a software system for research in algebraic geometry http://www.math.uiuc.edu/Macaulay2/ (Available at)

[6] B. Kim; R. Pandharipande The connectedness of the moduli space of maps to homogeneous spaces, Seoul, 2000, World Sci. Publ., River Edge, NJ, USA (2001), pp. 187-201

Cité par Sources :

Commentaires - Politique