On the existence of Ulrich vector bundles on some irregular surfaces

Lopez, Angelo

doi:10.1090/proc/15278

On the existence of Ulrich vector bundles on some irregular surfaces

Author: Angelo Felice Lopez
Journal: Proc. Amer. Math. Soc. 149 (2021), 13-26
MSC (2010): Primary 14J60; Secondary 14J27, 14J29
DOI: https://doi.org/10.1090/proc/15278
Published electronically: October 16, 2020
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We establish the existence of rank two Ulrich vector bundles on surfaces that are either of maximal Albanese dimension or with irregularity , under many embeddings. In particular, we get the first known examples of Ulrich vector bundles on irregular surfaces of general type. Another consequence is that every surface such that either $q \le 1$ or $q \ge 2$ and its minimal model has rank one, carries a simple rank two Ulrich vector bundle.

[A] I. V. Artamkin, Deformation of torsion-free sheaves on an algebraic surface, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 3, 435–468 (Russian); English transl., Math. USSR-Izv. 36 (1991), no. 3, 449–485. MR 1072690, https://doi.org/10.1070/IM1991v036n03ABEH002030
[ACM] 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, https://doi.org/10.1016/j.jpaa.2017.03.007
[AFO] 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, https://doi.org/10.1515/crelle-2014-0124
[B1] Arnaud Beauville, An introduction to Ulrich bundles, Eur. J. Math. 4 (2018), no. 1, 26–36. MR 3782216, https://doi.org/10.1007/s40879-017-0154-4
[B2] Arnaud Beauville, Ulrich bundles on abelian surfaces, Proc. Amer. Math. Soc. 144 (2016), no. 11, 4609–4611. MR 3544513, https://doi.org/10.1090/S0002-9939-2016-13091-8
[B3] A. Beauville. Ulrich bundles on surfaces with . Preprint arXiv:1607.00895 [math.AG].
[B4] Arnaud Beauville, Complex algebraic surfaces, 2nd ed., London Mathematical Society Student Texts, vol. 34, Cambridge University Press, Cambridge, 1996. Translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid. MR 1406314
[BN] Lev Borisov and Howard Nuer, Ulrich bundles on Enriques surfaces, Int. Math. Res. Not. IMRN 13 (2018), 4171–4189. MR 3829179, https://doi.org/10.1093/imrn/rnx018
[C1] Gianfranco Casnati, Special Ulrich bundles on non-special surfaces with 𝑝_{𝑔}=𝑞=0, Internat. J. Math. 28 (2017), no. 8, 1750061, 18. MR 3681122, https://doi.org/10.1142/S0129167X17500616
[C1e] Gianfranco Casnati, Erratum: “Special Ulrich bundles on non-special surfaces with 𝑝_{𝑔}=𝑞=0” [ MR3681122], Internat. J. Math. 29 (2018), no. 5, 1892001, 3. MR 3808056, https://doi.org/10.1142/S0129167X18920015
[C2] G. Casnati. Special Ulrich bundles on regular surfaces with non-negative Kodaira dimension. Preprint arXiv:1809.08565 [math.AG].
[C3] Gianfranco Casnati, Rank two stable Ulrich bundles on anticanonically embedded surfaces, Bull. Aust. Math. Soc. 95 (2017), no. 1, 22–37. MR 3592541, https://doi.org/10.1017/S0004972716000952
[C4] Gianfranco Casnati, Ulrich bundles on non-special surfaces with 𝑝_{𝑔}=0 and 𝑞=1, Rev. Mat. Complut. 32 (2019), no. 2, 559–574. MR 3942927, https://doi.org/10.1007/s13163-017-0248-z
[C4e] Gianfranco Casnati, Correction to: Ulrich bundles on non-special surfaces with 𝑝_{𝑔}=0 and 𝑞=1 [ MR3942927], Rev. Mat. Complut. 32 (2019), no. 2, 575–577. MR 3942928, https://doi.org/10.1007/s13163-019-00295-1
[CaHa] Marta Casanellas, Robin Hartshorne, Florian Geiss, and Frank-Olaf Schreyer, Stable Ulrich bundles, Internat. J. Math. 23 (2012), no. 8, 1250083, 50. MR 2949221, https://doi.org/10.1142/S0129167X12500838
[CKM] Emre Coskun, Rajesh S. Kulkarni, and Yusuf Mustopa, Pfaffian quartic surfaces and representations of Clifford algebras, Doc. Math. 17 (2012), 1003–1028. MR 3007683
[CoHu] E. Coskun, J. Huizenga. Brill-Noether Theorems, Ulrich bundles and the cohomology of moduli spaces of sheaves. Preliminary version, for the Proceedings of the ICM Satellite Conference on Moduli Spaces in Algebraic Geometry and Applications. Available at author's webpage.
[ES] David Eisenbud, Frank-Olaf Schreyer, and Jerzy Weyman, Resultants and Chow forms via exterior syzygies, J. Amer. Math. Soc. 16 (2003), no. 3, 537–579. MR 1969204, https://doi.org/10.1090/S0894-0347-03-00423-5
[F] Daniele Faenzi, Ulrich bundles on K3 surfaces, Algebra Number Theory 13 (2019), no. 6, 1443–1454. MR 3994571, https://doi.org/10.2140/ant.2019.13.1443
[GL] Mark Green and Robert Lazarsfeld, Deformation theory, generic vanishing theorems, and some conjectures of Enriques, Catanese and Beauville, Invent. Math. 90 (1987), no. 2, 389–407. MR 910207, https://doi.org/10.1007/BF01388711
[K] Yeongrak Kim, Ulrich bundles on blowing ups, C. R. Math. Acad. Sci. Paris 354 (2016), no. 12, 1215–1218 (English, with English and French summaries). MR 3573932, https://doi.org/10.1016/j.crma.2016.10.022
[KMM] Yujiro Kawamata, Katsumi Matsuda, and Kenji Matsuki, Introduction to the minimal model problem, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 283–360. MR 946243, https://doi.org/10.2969/aspm/01010283
[H] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
[Ha] Heinrich Hartmann, Cusps of the Kähler moduli space and stability conditions on K3 surfaces, Math. Ann. 354 (2012), no. 1, 1–42. MR 2957617, https://doi.org/10.1007/s00208-011-0719-3
[Hc] Christopher D. Hacon, A derived category approach to generic vanishing, J. Reine Angew. Math. 575 (2004), 173–187. MR 2097552, https://doi.org/10.1515/crll.2004.078
[HL] Daniel Huybrechts and Manfred Lehn, The geometry of moduli spaces of sheaves, 2nd ed., Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2010. MR 2665168
[HUB] 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, https://doi.org/10.1016/0022-4049(91)90147-T
[LMS] Angelo Felice Lopez, Roberto Muñoz, and José Carlos Sierra, On the extendability of elliptic surfaces of rank two and higher, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 1, 311–346 (English, with English and French summaries). MR 2514867
[Mi] Rick Miranda, The basic theory of elliptic surfaces, Dottorato di Ricerca in Matematica. [Doctorate in Mathematical Research], ETS Editrice, Pisa, 1989. MR 1078016
[MLP] Margarida Mendes Lopes and Rita Pardini, The geography of irregular surfaces, Current developments in algebraic geometry, Math. Sci. Res. Inst. Publ., vol. 59, Cambridge Univ. Press, Cambridge, 2012, pp. 349–378. MR 2931875
[MP] R. M. Miró-Roig and J. Pons-Llopis, Special Ulrich bundles on regular Weierstrass fibrations, Math. Z. 293 (2019), no. 3-4, 1431–1441. MR 4024593, https://doi.org/10.1007/s00209-019-02277-x
[Mu] Shigeru Mukai, Symplectic structure of the moduli space of sheaves on an abelian or 𝐾3 surface, Invent. Math. 77 (1984), no. 1, 101–116. MR 751133, https://doi.org/10.1007/BF01389137
[PT] Joan Pons-Llopis and Fabio Tonini, ACM bundles on del Pezzo surfaces, Matematiche (Catania) 64 (2009), no. 2, 177–211. MR 2800010
[S] Saverio Andrea Secci, On the existence of Ulrich bundles on blown-up varieties at a point, Boll. Unione Mat. Ital. 13 (2020), no. 1, 131–135. MR 4060515, https://doi.org/10.1007/s40574-019-00208-6