Skip to main navigation menu Skip to main content Skip to site footer

Windows and moduli of bundles on a curve

Abstract

We survey some results regarding semi-orthogonal decompositions on moduli spaces of vector bundles on a curve, with an emphasis on those that use the machinery of windows. We briefly outline some of the main ideas that make windows a crucial tool for these results.

2020 Mathematics Subject Classification:

14F08, 14H60

Keywords

moduli spaces, derived category, semi-orthogonal decompositions

Full text

Author Details

Sebastián Torres

Universidad Técnica Federico Santa María
Avenida España 1680, Office F323
Valparaíso, Chile
e-mail: sebastian.torresk@usm.cl


References

  1. M. Ballard, D. Favero, L. Katzarkov. Variation of geometric invariant theory quotients and derived categories. J. Reine Angew. Math. 746 (2019), 235–303.
  2. P. Belmans. Semiorthogonal decompositions for moduli of sheaves on curves. In: Interactions between Algebraic Geometry and Noncommutative Algebra. Oberwolfach Reports 15, 2 (2018), 1473–1476.
  3. P. Belmans. Seshadri's desingularisation in the Hodge diamond cutter, and a bold proposal, 2021, https://pbelmans.ncag.info/blog/2021/03/21/seshadris-desingularisation-in-hodge-diamond-cutter/.
  4. P. Belmans, J. Bose, S. Frei, B. Gould, J. Hotchkiss, A. Lamarche, J. Petok, C. Rodriguez Avila, S. Shah. On decompositions for Fano schemes of intersections of two quadrics, arXiv:2403.12517 [math.AG], 2024, https://arxiv.org/abs/2403.12517.
  5. P. Belmans, S. Galkin, S. Mukhopadhyay. Decompositions of moduli spaces of vector bundles and graph potentials. Forum Math. Sigma 11 (2023), Paper No. e16, 28 pp.
  6. P. Belmans, S. Mukhopadhyay. Admissible subcategories in derived categories of moduli of vector bundles on curves. Adv. Math. 351 (2019), 653–675.
  7. A. Bondal, D. Orlov. Semiorthogonal decomposition for algebraic varieties, arXiv:alg-geom/9506012, 1995, https://arxiv.org/abs/alg-geom/9506012.
  8. S. Del Baño. On the motive of moduli spaces of rank two vector bundles over a curve. Compositio Math. 131, 1 (2002), 1–30.
  9. A. V. Fonarev. Derived category of moduli of parabolic bundles on P1. Uspekhi Mat. Nauk 78, 3 (2023), 177–178 (in Russian); English translation Russian Math. Surveys 78, 3 (2023), 563–565.
  10. A. Fonarev, A. Kuznetsov. Derived categories of curves as components of Fano manifolds. J. Lond. Math. Soc. (2) 97, 1 (2018), 24–46.
  11. T. L. Gómez, K.-S. Lee. Motivic decompositions of moduli spaces of vector bundles on curves, arXiv:2007.06067 [math.AG], 2020, https://arxiv.org/abs/2007.06067.
  12. D. Halpern-Leistner. The derived category of a GIT quotient. J. Amer. Math. Soc. 28, 3 (2015), 871–912.
  13. K.-S. Lee. Remarks on motives of moduli spaces of rank 2 vector bundles on curves, arXiv:1806.11101 [math.AG], 2018, https://arxiv.org/abs/1806.11101.
  14. K.-S. Lee, H.-B. Moon. Derived category and ACM bundles of moduli space of vector bundles on a curve. Forum Math. Sigma 11 (2023), Paper No. e81, 23 pp.
  15. K.-S. Lee, H.-B. Moon. Derived categories of symmetric products and moduli spaces of vector bundles on a curve, arXiv:2309.15412 [math.AG], 2023, https://arxiv.org/abs/2309.15412.
  16. K.-S. Lee, M. S. Narasimhan. Symmetric products and moduli spaces of vector bundles of curves, arXiv:2106.04872 [math.AG], 2021, https://arxiv.org/abs/2106.04872.
  17. X. Lin. On nonexistence of semi-orthogonal decompositions in algebraic geometry, arXiv:2107.09564 [math.AG], 2021, https://arxiv.org/abs/2107.09564.
  18. M. S. Narasimhan. Derived categories of moduli spaces of vector bundles on curves II. In: Geometry, algebra, number theory, and their information technology applications (Eds A. Akbary, S. Gun), Springer Proc. Math. Stat., vol. 251, Cham, Springer, 2018, 375–382.
  19. S. Ramanan. The moduli spaces of vector bundles over an algebraic curve. Math. Ann. 200 (1973), 69–84.
  20. C. S. Seshadri. Moduli of vector bundles on curves with parabolic structures. Bull. Amer. Math. Soc. 83, 1 (1977), 124–126.
  21. E. Sink, J. Tevelev. Noncommutative resolution of SUC(2), arXiv:2405.08891 [math.AG], 2024, https://arxiv.org/abs/2405.08891.
  22. C. Teleman. The quantization conjecture revisited. Ann. of Math. (2) 152, 1 (2000), 1–43.
  23. J. Tevelev. Braid and Phantom, arXiv:2304.01825 [math.AG], 2023, https://arxiv.org/abs/2304.01825.
  24. J. Tevelev, S. Torres. The BGMN conjecture via stable pairs. Duke Math. J. 173, 18 (2024), 3495–3557.
  25. M. Thaddeus. Stable pairs, linear systems and the Verlinde formula. Invent. Math. 117, 2 (1994), 317–353.
  26. K. Xu. Moduli of Vector Bundles on Curve and Semiorthogonal Decomposition. Pro-Quest LLC, Ann Arbor, MI, 2023. Thesis (Ph.D.)–Harvard University.
  27. K. Xu, S.-T. Yau. Semiorthogonal decomposition of Db(BunL2), arXiv:2108.13353 [math.AG], 2021 https://arxiv.org/abs/2108.13353.