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, 14H60Keywords
moduli spaces, derived category, semi-orthogonal decompositions
Author Details
Sebastián Torres
Universidad Técnica Federico Santa María
Avenida España 1680, Office F323
Valparaíso, Chile
e-mail: sebastian.torresk@usm.cl
References
- M. Ballard, D. Favero, L. Katzarkov. Variation of geometric invariant theory quotients and derived categories. J. Reine Angew. Math. 746 (2019), 235–303.
- 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.
- 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/.
- 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.
- 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.
- P. Belmans, S. Mukhopadhyay. Admissible subcategories in derived categories of moduli of vector bundles on curves. Adv. Math. 351 (2019), 653–675.
- A. Bondal, D. Orlov. Semiorthogonal decomposition for algebraic varieties, arXiv:alg-geom/9506012, 1995, https://arxiv.org/abs/alg-geom/9506012.
- S. Del Baño. On the motive of moduli spaces of rank two vector bundles over a curve. Compositio Math. 131, 1 (2002), 1–30.
- 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.
- A. Fonarev, A. Kuznetsov. Derived categories of curves as components of Fano manifolds. J. Lond. Math. Soc. (2) 97, 1 (2018), 24–46.
- 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.
- D. Halpern-Leistner. The derived category of a GIT quotient. J. Amer. Math. Soc. 28, 3 (2015), 871–912.
- 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.
- 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.
- 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.
- 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.
- X. Lin. On nonexistence of semi-orthogonal decompositions in algebraic geometry, arXiv:2107.09564 [math.AG], 2021, https://arxiv.org/abs/2107.09564.
- 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.
- S. Ramanan. The moduli spaces of vector bundles over an algebraic curve. Math. Ann. 200 (1973), 69–84.
- C. S. Seshadri. Moduli of vector bundles on curves with parabolic structures. Bull. Amer. Math. Soc. 83, 1 (1977), 124–126.
- E. Sink, J. Tevelev. Noncommutative resolution of SUC(2), arXiv:2405.08891 [math.AG], 2024, https://arxiv.org/abs/2405.08891.
- C. Teleman. The quantization conjecture revisited. Ann. of Math. (2) 152, 1 (2000), 1–43.
- J. Tevelev. Braid and Phantom, arXiv:2304.01825 [math.AG], 2023, https://arxiv.org/abs/2304.01825.
- J. Tevelev, S. Torres. The BGMN conjecture via stable pairs. Duke Math. J. 173, 18 (2024), 3495–3557.
- M. Thaddeus. Stable pairs, linear systems and the Verlinde formula. Invent. Math. 117, 2 (1994), 317–353.
- K. Xu. Moduli of Vector Bundles on Curve and Semiorthogonal Decomposition. Pro-Quest LLC, Ann Arbor, MI, 2023. Thesis (Ph.D.)–Harvard University.
- K. Xu, S.-T. Yau. Semiorthogonal decomposition of Db(BunL2), arXiv:2108.13353 [math.AG], 2021 https://arxiv.org/abs/2108.13353.