Cubic symmetroids and vector bundles on a quadric surface

Sukmoon Huh

doi:10.1016/j.crma.2013.07.018

Algebraic Geometry

Cubic symmetroids and vector bundles on a quadric surface

Presented by: Claire Voisin

Sukmoon Huh ¹

¹ Department of Mathematics, Sungkyunkwan University, 300 Cheoncheon-dong, Suwon 440-746, Republic of Korea

Comptes Rendus. Mathématique, Volume 351 (2013) no. 13-14, pp. 557-560.

Abstract
Résumé

We investigate the jumping conics of stable vector bundles $E$ of rank 2 on a smooth quadric surface Q with the Chern classes $c_{1} = O_{Q} (- 1, - 1)$ and $c_{2} = 4$ with respect to the ample line bundle $O_{Q} (1, 1)$ . As a consequence, we prove that the set of jumping conics $S (E)$ uniquely determines $E$ and that the moduli space of such bundles is rational.

Nous étudions les coniques de saut des fibrés vectoriels stables $E$ de rang 2 sur une surface quadratique lisse Q de classes de Chern $c_{1} = O_{Q} (- 1, - 1)$ et $c_{2} = 4$ relativement au fibré en droites ample $O_{Q} (1, 1)$ . Nous en déduisons que lʼensemble des coniques de saut $S (E)$ détermine $E$ de maniére unique et que lʼespace de modules de ce type de fibrés est rationnel.

Received: 2013-04-26
Accepted: 2013-07-23
Published online: 2013-07-01

DOI: 10.1016/j.crma.2013.07.018

Author's affiliations:

Sukmoon Huh ¹

¹ Department of Mathematics, Sungkyunkwan University, 300 Cheoncheon-dong, Suwon 440-746, Republic of Korea

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     title = {Cubic symmetroids and vector bundles on a quadric surface},
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Sukmoon Huh. Cubic symmetroids and vector bundles on a quadric surface. Comptes Rendus. Mathématique, Volume 351 (2013) no. 13-14, pp. 557-560. doi : 10.1016/j.crma.2013.07.018. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.07.018/

References
Cited by

[1] F. Catanese Footnotes to a theorem of I. Reider, LʼAquila, 1988 (Lect. Notes Math.), Volume vol. 1417, Springer, Berlin (1990), pp. 67-74

[2] L. Costa; R.M. Miro-Ŕoig Rationality of moduli spaces of vector bundles on rational surfaces, Nagoya Math. J., Volume 162 (2002), pp. 43-69

[3] I. Dolgachev Classical Algebraic Geometry: A Modern View, Cambridge University Press, Cambridge, UK, 2012 (xii+639 pp)

[4] D. Gieseker On the moduli of vector bundles on an algebraic surface, Ann. of Math. (2), Volume 106 (1977) no. 1, pp. 45-60

[5] R. Hartshorne Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, Springer-Verlag, New York, 1977

[6] S. Huh Jumping conics on a smooth quadric in $P_{3}$ , Ann. Mat. Pura Appl. (4), Volume 190 (2011) no. 2, pp. 195-208

[7] S. Huh Moduli of stable sheaves on a smooth quadric and a Brill–Noether locus, J. Pure Appl. Algebra, Volume 215 (2011) no. 9, pp. 2099-2105

[8] J. Le Potier Lectures on Vector Bundles, Cambridge Studies in Advanced Mathematics, vol. 54, Cambridge University Press, Cambridge, 1997 (Translated by A. Maciocia)

[9] H. Nakajima Lectures on Hilbert Schemes of Points on Surfaces, University Lecture Series, vol. 18, American Mathematical Society, Providence, RI, 1999

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