Abstract
We obtain an upper bound on the first Chern class and the Castelnuovo-Mumford regularity of an initialized rank 3 ACM bundle on a general hypersurface in \(\mathbb {P}^4.\) As a corollary, we prove that a general hypersurface in \(\mathbb {P}^4\) of degree \(d \ge 4\) does not support a rank 3 Ulrich bundle. We also make progress on the base case of a generic version of a conjecture by Buchweitz, Greuel and Schreyer.
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Notes
See discussion in Sect. 2.
References
Beauville, A.: An introduction to Ulrich bundles. Eur. J. Math. 4(1), 26–36 (2018)
Beauville, A.: Determinantal hypersurfaces. Michigan Math. J. 48, 39–64 (2000)
Buchweitz, R.-O., Greuel, G.-M., Schreyer, F.-O.: Cohen-Macaulay modules on hypersurface singularities II. Invent. Math. 88, 165–182 (1987)
Eisenbud, D.: Homological algebra on a complete intersection, with an application to group representations. Trans. Amer. Math. Soc. 260(1), 35–64 (1980)
Eisenbud, D., Schreyer, F.-O., Weyman, J.: Resultants and Chow forms via exterior syzygies. J. Amer. Math. Soc. 16(3), 537–579 (2003)
Hartshorne, R.: Algebraic geometry, Graduate Texts in Mathematics, No. 52, p. xvi+496. Springer-Verlag, New York-Heidelberg (1977)
Horrocks, G.: Vector bundles on the punctured spectrum of a local ring. Proc. London Math. Soc. (3) 14, 689–713 (1964)
Lahoz, M., Macrì, E., Stellari, P.: Arithmetically Cohen-Macaulay bundles on cubic threefolds. Algebr. Geom. 2(2), 231–269 (2015)
Madonna, C.: A splitting criterion for rank 2 vector bundles on hypersurfaces in \({\bf P}^4\). Rend. Sem. Mat. Univ. Politec. Torino 56(2), 43–54 (1998)
Mohan Kumar, N., Rao, A.P., Ravindra, G.V.: Arithmetically Cohen-Macaulay bundles on hypersurfaces. Comment. Math Helvetici 82(4), 829–843 (2007)
Mohan Kumar, N., Rao, A.P., Ravindra, G.V.: Arithmetically Cohen-Macaulay bundles on three dimensional hypersurfaces. International Math Research Notices, (8), Art. ID rnm025, p. 11 (2007)
Ravindra, G.V.: Curves on threefolds and a conjecture of Griffiths-Harris. Math. Annalen 345(3), 731–748 (2009)
Ravindra, G.V., Tripathi, A.: Extensions of vector bundles with application to Noether-Lefschetz theorems. Commun. Contemp. Math. 15(5), 1350003–1350020 (2013)
Ravindra, Girivaru V., Tripathi, Amit: Remarks on higher-rank ACM bundles on hypersurfaces. C. R. Math. Acad. Sci. Paris 356(11–12), 1215–1221 (2018)
Ravindra, G.V., Tripathi, A.: Rank 3 ACM bundles on general hypersurfaces in \(\mathbb{P}^5\). Adv. Math. 355, 106780 (2019)
Tripathi, A.: Rank 3 arithmetically Cohen-Macaulay bundles over hypersurfaces. J. Algebra 478, 1–11 (2017)
Acknowledgements
The authors would like to thank the referee for many comments and suggestions. The first author was partially supported by a grant from the Simons Foundation (Award ID:830817). The second author was partially supported by the science and engineering research board (SERB) grant MTR/2020/000164.
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Ravindra, G.V., Tripathi, A. On the base case of a conjecture on ACM bundles over hypersurfaces. Geom Dedicata 216, 49 (2022). https://doi.org/10.1007/s10711-022-00711-9
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DOI: https://doi.org/10.1007/s10711-022-00711-9