Remarks on higher-rank ACM bundles on hypersurfaces

Ravindra, Girivaru V.; Tripathi, Amit

doi:10.1016/j.crma.2018.10.004

Algebraic geometry

Remarks on higher-rank ACM bundles on hypersurfaces
[Remarques sur les fibrés ACM de rang supérieur sur les hypersurfaces]

Présenté par : Claire Voisin

Ravindra, Girivaru V.¹ ; Tripathi, Amit 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, Tome 356 (2018) no. 11-12, pp. 1215-1221.

Résumé
Abstract

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.

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.

Reçu le : 2018-06-18
Accepté le : 2018-10-22
Publié le : 2018-10-25

DOI : 10.1016/j.crma.2018.10.004

Affiliations des auteurs :

Ravindra, Girivaru V. ¹ ; Tripathi, Amit ²

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Ravindra, Girivaru V.; Tripathi, Amit. Remarks on higher-rank ACM bundles on hypersurfaces. Comptes Rendus. Mathématique, Tome 356 (2018) no. 11-12, pp. 1215-1221. doi : 10.1016/j.crma.2018.10.004. http://www.numdam.org/articles/10.1016/j.crma.2018.10.004/

Bibliographie
Cité par

[1] Beauville, A. Determinantal hypersurfaces, Dedicated to William Fulton on the occasion of his 60th birthday, Mich. Math. J., Volume 48 (2000), pp. 39-64

[2] Biswas, J.; Ravindra, G.V. Arithmetically Cohen–Macaulay bundles on complete intersection varieties of sufficiently high multi-degree, Math. Z., Volume 265 (2010) no. 3, pp. 493-509

[3] Buchweitz, R.-O.; Greuel, G.-M.; Schreyer, F.-O. Cohen–Macaulay modules on hypersurface singularities II, Invent. Math., Volume 88 (1987), pp. 165-182

[4] Chiantini, L.; Madonna, C.K. ACM bundles on general hypersurfaces in $P^{5}$ of low degree, Collect. Math., Volume 56 (2005) no. 1, pp. 85-96

[5] Faenzi, D. Some Applications of Vector Bundles in Algebraic Geometry, 2013 (Habilitation à diriger des recherches)

[6] Horrocks, G. Vector bundles on the punctured spectrum of a local ring, Proc. Lond. Math. Soc. (3), Volume 14 (1964), pp. 689-713

[7] Kleppe, H. Deformation of schemes defined by vanishing of Pfaffians, J. Algebra, Volume 53 (1978) no. 1, pp. 84-92

[8] Knörrer, H. Cohen–Macaulay modules on hypersurface singularities, I, Invent. Math., Volume 88 (1987) no. 1, pp. 153-164

[9] Mohan Kumar, N.; Rao, A.P.; Ravindra, G.V. Arithmetically Cohen–Macaulay bundles on hypersurfaces, Comment. Math. Helv., Volume 82 (2007) no. 4, pp. 829-843

[10] Mohan Kumar, N.; Rao, A.P.; Ravindra, G.V. Arithmetically Cohen–Macaulay bundles on three dimensional hypersurfaces, Int. Math. Res. Not., Volume 2007 (2007) no. 8

[11] Mohan Kumar, N.; Rao, A.P.; Ravindra, G.V. On codimension two subvarieties of hypersurfaces, Motives and Algebraic Cycles, Fields Inst. Commun., vol. 56, Amer. Math. Soc., Providence, RI, 2009, pp. 167-174

[12] Ravindra, G.V. Curves on threefolds and a conjecture of Griffiths–Harris, Math. Ann., Volume 345 (2009) no. 3, pp. 731-748

[13] G.V. Ravindra, Topics in the geometry of vector bundles, Proposal submitted to the NSF, 2009.

[14] Ravindra, G.V.; Tripathi, A. Extensions of vector bundles with application to Noether–Lefschetz theorems, Commun. Contemp. Math., Volume 15 (2013) no. 5

[15] Ravindra, G.V.; Tripathi, A. Torsion points and matrices defining elliptic curves, Int. J. Algebra Comput., Volume 24 (2014) no. 06, pp. 879-891

[16] Tripathi, A. Splitting of low-rank ACM bundles on hypersurfaces of high dimension, Commun. Algebra, Volume 44 (2016) no. 3, pp. 1011-1017

[17] Tripathi, A. Rank 3 arithmetically Cohen–Macaulay bundles over hypersurfaces, J. Algebra, Volume 478 (2017), pp. 1-11

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