Comptes Rendus
no. 10
Algebraic geometry
Remarks on minimal rational curves on moduli spaces of stable bundles
Claire Voisin
Min Liu1
1School of Mathematics and Statistics, Qingdao University, Qingdao 266071, PR China
Comptes Rendus. Mathématique, Volume 354 (2016) no. 10, pp. 1013-1017

Let C be a smooth projective curve of genus g≥2 over an algebraically closed field of characteristic zero, and M be the moduli space of stable bundles of rank 2 and with fixed determinant L of degree d on the curve C. When g=3 and d is even, we prove that, for any point [W]∈M, there is a minimal rational curve passing through [W], which is not a Hecke curve. This complements a theorem of Xiaotao Sun.

Soient C une courbe projective lisse de genre g≥2 et M l'espace des modules de faisceaux stables de rang 2 et de déterminant fixe L de degré d sur C. Nous prouvons que, lorsque g=3 et d est pair, il existe, pour tout point [W]∈M, une courbe rationnelle minimale passant par [W], qui n'est pas une courbe de Hecke. Cela complète un théorème de Xiaotao Sun.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2016.08.007

Min Liu  1

1 School of Mathematics and Statistics, Qingdao University, Qingdao 266071, PR China 
Min Liu. Remarks on minimal rational curves on moduli spaces of stable bundles. Comptes Rendus. Mathématique, Volume 354 (2016) no. 10, pp. 1013-1017. doi: 10.1016/j.crma.2016.08.007
@article{CRMATH_2016__354_10_1013_0,
     author = {Min Liu},
     title = {Remarks on minimal rational curves on moduli spaces of stable bundles},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1013--1017},
     year = {2016},
     publisher = {Elsevier},
     volume = {354},
     number = {10},
     doi = {10.1016/j.crma.2016.08.007},
     language = {en},
}
TY  - JOUR
AU  - Min Liu
TI  - Remarks on minimal rational curves on moduli spaces of stable bundles
JO  - Comptes Rendus. Mathématique
PY  - 2016
SP  - 1013
EP  - 1017
VL  - 354
IS  - 10
PB  - Elsevier
DO  - 10.1016/j.crma.2016.08.007
LA  - en
ID  - CRMATH_2016__354_10_1013_0
ER  - 
%0 Journal Article
%A Min Liu
%T Remarks on minimal rational curves on moduli spaces of stable bundles
%J Comptes Rendus. Mathématique
%D 2016
%P 1013-1017
%V 354
%N 10
%I Elsevier
%R 10.1016/j.crma.2016.08.007
%G en
%F CRMATH_2016__354_10_1013_0

[1] J.-M. Drezet; M.S. Narasimhan Groupe de Picard des variétés de modules de fibrés semistables sur les courbes algébriques, Invent. Math., Volume 97 (1989), pp. 53-94

[2] H. Lang; M.S. Narasimhan Maximal subbundles of rank two vector bundles on curves, Math. Ann., Volume 266 (1983), pp. 55-72

[3] M. Liu Small rational curves on the moduli space of stable bundles, Int. J. Math., Volume 23 (2012) no. 8

[4] N. Mok; X. Sun Remarks on lines and minimal rational curves, Sci. China Ser. A, Math., Volume 52 (2009) no. 4, pp. 617-630

[5] N. Mok; X. Sun Erratum to: Remarks on lines and minimal rational curves, Sci. China Ser. A, Math., Volume 5 (2014) no. 9, p. 1992

[6] M. Nagata On self intersection number of vector bundles of rank 2 on a Riemann surface, Nagoya Math. J., Volume 37 (1970), pp. 191-196

[7] M.S. Narasimhan; S. Ramanan Moduli of vector bundles on a compact Riemann surface, Ann. of Math. (2), Volume 89 (1969), pp. 14-51

[8] M.S. Narasimhan; S. Ramanan Geometry of Hecke cycles I, C.P. Ramanujam – A Tribute, Tata Institute, Springer Verlag, 1978, pp. 291-345

[9] S. Ramanan The moduli spaces of vector bundles over an algebraic curve, Math. Ann., Volume 200 (1973), pp. 69-84

[10] X. Sun Minimal rational curves on the moduli spaces of stable bundles, Math. Ann., Volume 331 (2005), pp. 925-937

[11] X. Sun Elliptic curves in moduli spaces of stable bundles, Pure Appl. Math. Q., Volume 7 (2011) no. 4

Cited by Sources:

☆ Supported by the National Natural Science Foundation of China (Grant No. 11401330).

Comments - Policy