Remarks on higher-rank ACM bundles on hypersurfaces

Girivaru V. Ravindra; Amit Tripathi

doi:10.1016/j.crma.2018.10.004

Algebraic geometry

Remarks on higher-rank ACM bundles on hypersurfaces

Presented by: Claire Voisin

Girivaru V. Ravindra ¹; Amit Tripathi ²

¹ Department of Mathematics, University of Missouri – St. Louis, St. Louis, MO 63121, USA
² Department of Mathematics, Indian Institute of Technology Hyderabad, Kandi, Sangareddy, Hyderabad – 502285, Telangana, India

Comptes Rendus. Mathématique, Volume 356 (2018) no. 11-12, pp. 1215-1221.

Abstract
Résumé

In terms of the number of generators, one of the simplest non-split rank-3 arithmetically Cohen–Macaulay bundles on a smooth hypersurface in $P^{5}$ is 6-generated. We prove that a general hypersurface in $P^{5}$ of degree $d \geq 3$ does not support such a bundle. We also prove that a smooth positive dimensional hypersurface in projective space of even degree does not support an Ulrich bundle of odd rank and determinant of the form $O_{X} (c)$ for some integer c. This verifies some cases of conjectures we discuss here.

En termes de nombre de générateurs, le fibré de rang 3 arithmétiquement Cohen–Macaulay, non décomposé, le plus simple sur une hypersurface de $P^{5}$ , est engendré en rang 6. Nous montrons qu'une hypersurface générale dans $P^{5}$ , de degré $d \geq 3$ , n'admet pas un tel fibré. Nous montrons également qu'une hypersurface lisse de dimension positive dans un espace projectif, de degré pair, n'admet pas de faisceau d'Ulrich de rang impair. Ceci permet de vérifier quelques cas de conjectures, que nous discutons ici.

Received: 2018-06-18
Accepted: 2018-10-22
Published online: 2018-10-25

DOI: 10.1016/j.crma.2018.10.004

Author's affiliations:

Girivaru V. Ravindra ¹; Amit Tripathi ²

¹ Department of Mathematics, University of Missouri – St. Louis, St. Louis, MO 63121, USA
² Department of Mathematics, Indian Institute of Technology Hyderabad, Kandi, Sangareddy, Hyderabad – 502285, Telangana, India

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     title = {Remarks on higher-rank {ACM} bundles on hypersurfaces},
     journal = {Comptes Rendus. Math\'ematique},
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     year = {2018},
     doi = {10.1016/j.crma.2018.10.004},
     language = {en},
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Girivaru V. Ravindra; Amit Tripathi. Remarks on higher-rank ACM bundles on hypersurfaces. Comptes Rendus. Mathématique, Volume 356 (2018) no. 11-12, pp. 1215-1221. doi : 10.1016/j.crma.2018.10.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.10.004/

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