Comptes Rendus
no. 10
Algebraic geometry
Remarks on minimal rational curves on moduli spaces of stable bundles

Presented by: Claire Voisin

Min Liu1
1 School 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 
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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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.08.007/

[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

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

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