Ulrich bundles on Brauer–Severi varieties

Novaković, Saša

doi:10.1090/proc/14723

Ulrich bundles on Brauer–Severi varieties
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by Saša Novaković

Proc. Amer. Math. Soc. 152 (2024), 7-22

DOI: https://doi.org/10.1090/proc/14723

Published electronically: October 16, 2023

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Abstract:

We prove the existence of Ulrich bundles on any Brauer–Severi variety. In some cases, the minimal possible rank of the obtained Ulrich bundles equals the period of the Brauer–Severi variety. Moreover, we find a formula for the rank of an Ulrich bundle involving the period of the considered Brauer–Severi variety $X$, at least if $\mathrm {dim}(X)=p-1$ for an odd prime $p$. This formula implies that the rank of any Ulrich bundle on such a Brauer–Severi variety $X$ must be a multiple of the period.

References

Marian Aprodu, Laura Costa, and Rosa Maria Miró-Roig, Ulrich bundles on ruled surfaces, J. Pure Appl. Algebra 222 (2018), no. 1, 131–138. MR 3680998, DOI 10.1016/j.jpaa.2017.03.007
Marian Aprodu, Gavril Farkas, and Angela Ortega, Minimal resolutions, Chow forms and Ulrich bundles on $K3$ surfaces, J. Reine Angew. Math. 730 (2017), 225–249. MR 3692019, DOI 10.1515/crelle-2014-0124
Lucian Bădescu, Infinitesimal extensions of rank two vector bundles on submanifolds of small codimension, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 58(106) (2015), no. 3, 231–243. MR 3410253
Arnaud Beauville, Ulrich bundles on abelian surfaces, Proc. Amer. Math. Soc. 144 (2016), no. 11, 4609–4611. MR 3544513, DOI 10.1090/proc/13091
Arnaud Beauville, An introduction to Ulrich bundles, Eur. J. Math. 4 (2018), no. 1, 26–36. MR 3782216, DOI 10.1007/s40879-017-0154-4
Marcello Bernardara, A semiorthogonal decomposition for Brauer-Severi schemes, Math. Nachr. 282 (2009), no. 10, 1406–1413. MR 2571702, DOI 10.1002/mana.200610826
Joseph P. Brennan, Jürgen Herzog, and Bernd Ulrich, Maximally generated Cohen-Macaulay modules, Math. Scand. 61 (1987), no. 2, 181–203. MR 947472, DOI 10.7146/math.scand.a-12198
Lev Borisov and Howard Nuer, Ulrich bundles on Enriques surfaces, Int. Math. Res. Not. IMRN 13 (2018), 4171–4189. MR 3829179, DOI 10.1093/imrn/rnx018
I. Coskun, J. Huizenga and M. Woolf: Equivariant Ulrich bundles on Flag varieties, arXiv:1507.00102v2 [math.AG] (2015).
Laura Costa, Rosa M. Miró-Roig, and Joan Pons-Llopis, The representation type of Segre varieties, Adv. Math. 230 (2012), no. 4-6, 1995–2013. MR 2927362, DOI 10.1016/j.aim.2012.03.034
L. Costa and R. M. Miró-Roig, $GL(V)$-invariant Ulrich bundles on Grassmannians, Math. Ann. 361 (2015), no. 1-2, 443–457. MR 3302625, DOI 10.1007/s00208-014-1076-9
L. Costa and R. M. Miró-Roig, Cohomological characterization of vector bundles on multiprojective spaces, J. Algebra 294 (2005), no. 1, 73–96. MR 2179715, DOI 10.1016/j.jalgebra.2005.08.035
Aise Johan de Jong and Wei Ho, Genus one curves and Brauer-Severi varieties, Math. Res. Lett. 19 (2012), no. 6, 1357–1359. MR 3091612, DOI 10.4310/MRL.2012.v19.n6.a13
David Eisenbud and Frank-Olaf Schreyer, Resultants and Chow forms via exterior syzygies, J. Amer. Math. Soc. 16 (2003), no. 3, 537–579. With an appendix by Jerzy Weyman. MR 1969204, DOI 10.1090/S0894-0347-03-00423-5
Philippe Gille and Tamás Szamuely, Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics, vol. 101, Cambridge University Press, Cambridge, 2006. MR 2266528, DOI 10.1017/CBO9780511607219
A. L. Gorodentsev and S. A. Kuleshov, Helix theory, Mosc. Math. J. 4 (2004), no. 2, 377–440, 535 (English, with English and Russian summaries). MR 2108443, DOI 10.17323/1609-4514-2004-4-2-377-440
Douglas Allen Hanes, Special conditions on maximal Cohen-Macaulay modules, and applications to the theory of multiplicities, ProQuest LLC, Ann Arbor, MI, 1999. Thesis (Ph.D.)–University of Michigan. MR 2699070
J. Herzog, B. Ulrich, and J. Backelin, Linear maximal Cohen-Macaulay modules over strict complete intersections, J. Pure Appl. Algebra 71 (1991), no. 2-3, 187–202. MR 1117634, DOI 10.1016/0022-4049(91)90147-T
Christian Liedtke, Morphisms to Brauer-Severi varieties, with applications to del Pezzo surfaces, Geometry over nonclosed fields, Simons Symp., Springer, Cham, 2017, pp. 157–196. MR 3644253
Z. Lin, Ulrich bundles on projective spaces, arXiv:1703.06424v1 [math.AG], 2017.
Rosa M. Miró-Roig, The representation type of rational normal scrolls, Rend. Circ. Mat. Palermo (2) 62 (2013), no. 1, 153–164. MR 3031575, DOI 10.1007/s12215-013-0113-y
S. Novaković, Absolutely split locally free sheaves on proper $k$-schemes and Brauer–Severi varieties, arXiv:1501.00859v2 [math.AG], 2018.
Dmitri Orlov, Smooth and proper noncommutative schemes and gluing of DG categories, Adv. Math. 302 (2016), 59–105. MR 3545926, DOI 10.1016/j.aim.2016.07.014
David J. Saltman, Lectures on division algebras, CBMS Regional Conference Series in Mathematics, vol. 94, Published by American Mathematical Society, Providence, RI; on behalf of Conference Board of the Mathematical Sciences, Washington, DC, 1999. MR 1692654, DOI 10.1090/cbms/094