Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

REMARKS ON ULRICH BUNDLES OF SMALL RANKS OVER QUARTIC FOURFOLDS

  • Yeongrak Kim (Department of Mathematics & Institute of Mathematical Science Pusan National University)
  • Received : 2023.02.27
  • Accepted : 2023.12.07
  • Published : 2024.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we investigate a few strategies to construct Ulrich bundles of small ranks over smooth fourfolds in ℙ5, with a focus on the case of special quartic fourfolds. First, we give a necessary condition for Ulrich bundles over a very general quartic fourfold in terms of the rank and the Chern classes. Second, we give an equivalent condition for Pfaffian fourfolds in every degree in terms of arithmetically Gorenstein surfaces therein. Finally, we design a computer-based experiment to look for Ulrich bundles of small rank over special quartic fourfolds via deformation theory. This experiment gives a construction of numerically Ulrich sheaf of rank 4 over a random quartic fourfold containing a del Pezzo surface of degree 5.

Keywords

Acknowledgement

This work was supported by a 2-Year Research Grant of Pusan National University.

References

  1. J. Backelin and J. Herzog, On Ulrich-modules over hypersurface rings, in Commutative algebra (Berkeley, CA, 1987), 63-68, Math. Sci. Res. Inst. Publ., 15, Springer, New York, 1989. https://doi.org/10.1007/978-1-4612-3660-3_4 
  2. A. Beauville, Determinantal hypersurfaces, Michigan Math. J. 48 (2000), 39-64. https://doi.org/10.1307/mmj/1030132707 
  3. M. Blaser, D. Eisenbud, and F.-O. Schreyer, Ulrich complexity, Differential Geom. Appl. 55 (2017), 128-145. https://doi.org/10.1016/j.difgeo.2017.06.001 
  4. D. A. Buchsbaum and D. Eisenbud, Algebra structures for finite free resolutions, and some structure theorems for ideals of codimension 3, Amer. J. Math. 99 (1977), no. 3, 447-485. https://doi.org/10.2307/2373926 
  5. R.-O. Buchweitz, D. Eisenbud, and J. Herzog, Cohen-Macaulay modules on quadrics, in Singularities, representation of algebras, and vector bundles (Lambrecht, 1985), 58-116, Lecture Notes in Math., 1273, Springer, Berlin, 1987. https://doi.org/10.1007/BFb0078838 
  6. M. Casanellas, R. Hartshorne, F. Geiss, and F.-O. Schreyer, Stable Ulrich bundles, Internat. J. Math. 23 (2012), no. 8, 1250083, 50 pp. https://doi.org/10.1142/S0129167X12500838 
  7. A. C. Dixon, Note on the reduction of a ternary quantic to a symmetrical determinant, Proc. Cambridge Philos. Soc. 11 (1902), 350-351. 
  8. D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), no. 1, 35-64. https://doi.org/10.2307/1999875 
  9. D. Eisenbud and F.-O. Schreyer, Boij-Soderberg theory, in Combinatorial aspects of commutative algebra and algebraic geometry, 35-48, Abel Symp., 6, Springer, Berlin, 2011. https://doi.org/10.1007/978-3-642-19492-4_3 
  10. D. Eisenbud, F.-O. Schreyer, and J. Weyman, Resultants and Chow forms via exterior syzygies, J. Amer. Math. Soc. 16 (2003), no. 3, 537-579. https://doi.org/10.1090/S0894-0347-03-00423-5 
  11. D. Faenzi and Y. Kim, Ulrich bundles on cubic fourfolds, Comment. Math. Helv. 97 (2022), no. 4, 691-728. https://doi.org/10.4171/cmh/546 
  12. D. Grayson and M. Stillman, Macaulay2 - a software system for algebraic geometry and commutative algebra, available at: https://macaulay2.com 
  13. R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, No. 52, Springer, New York, 1977. https://doi.org/10.1007/978-1-4757-3849-0 
  14. A. Iliev and L. Manivel, On cubic hypersurfaces of dimensions 7 and 8, Proc. Lond. Math. Soc. (3) 108 (2014), no. 2, 517-540. https://doi.org/10.1112/plms/pdt042 
  15. Y. Kim, Macaulay2 scripts: "Remarks on Ulrich bundles on Pfaffian fourfolds, available at: https://sites.google.com/view/yeongrak/publications 
  16. Y. Kim and F.-O. Schreyer, An explicit matrix factorization of cubic hypersurfaces of small dimension, J. Pure Appl. Algebra 224 (2020), no. 8, 106346, 13 pp. https://doi.org/10.1016/j.jpaa.2020.106346 
  17. H. Knorrer, Cohen-Macaulay modules on hypersurface singularities. I, Invent. Math. 88 (1987), no. 1, 153-164. https://doi.org/10.1007/BF01405095 
  18. C. Lehn, M. Lehn, C. Sorger, and D. van Straten, Twisted cubics on cubic fourfolds, J. Reine Angew. Math. 731 (2017), 87-128. https://doi.org/10.1515/crelle-2014-0144 
  19. L. Ma, Lech's conjecture in dimension three, Adv. Math. 322 (2017), 940-970. https://doi.org/10.1016/j.aim.2017.10.032 
  20. L. Ma, Lim Ulrich sequences and Lech's conjecture, Invent. Math. 231 (2023), no. 1, 407-429. https://doi.org/10.1007/s00222-022-01149-2 
  21. L. Manivel, Ulrich and aCM bundles from invariant theory, Comm. Algebra 47 (2019), no. 2, 706-718. https://doi.org/10.1080/00927872.2018.1495222 
  22. G. V. Ravindra and A. Tripathi, Remarks on higher-rank ACM bundles on hypersurfaces, C. R. Math. Acad. Sci. Paris 356 (2018), no. 11-12, 1215-1221. https://doi.org/10.1016/j.crma.2018.10.004 
  23. J. Shamash, The Poincar'e series of a local ring, J. Algebra 12 (1969), 453-470. https://doi.org/10.1016/0021-8693(69)90023-4 
  24. R. P. Thomas, A holomorphic Casson invariant for Calabi-Yau 3-folds, and bundles on K3 fibrations, J. Differential Geom. 54 (2000), no. 2, 367-438. https://doi.org/10.4310/jdg/1214341649 
  25. H. L. Truong and H. N. Yen, Stable Ulrich bundles on cubic fourfolds, preprint, arXiv:2206.05285. 
  26. B. Ulrich, Gorenstein rings and modules with high numbers of generators, Math. Z. 188 (1984), no. 1, 23-32. https://doi.org/10.1007/BF01163869