Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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LOW RANK ORTHOGONAL BUNDLES AND QUADRIC FIBRATIONS

  • Insong Choe (Department of Mathematics Konkuk University) ;
  • George H. Hitching (Oslo Metropolitan University)
  • Received : 2022.03.13
  • Accepted : 2023.07.19
  • Published : 2023.11.01
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Abstract

Let C be a curve and V → C an orthogonal vector bundle of rank r. For r ≤ 6, the structure of V can be described using tensor, symmetric and exterior products of bundles of lower rank, essentially due to the existence of exceptional isomorphisms between Spin(r, ℂ) and other groups for these r. We analyze these structures in detail, and in particular use them to describe moduli spaces of orthogonal bundles. Furthermore, the locus of isotropic vectors in V defines a quadric subfibration QV ⊂ ℙV . Using familiar results on quadrics of low dimension, we exhibit isomorphisms between isotropic Quot schemes of V and certain ordinary Quot schemes of line subbundles. In particular, for r ≤ 6 this gives a method for enumerating the isotropic subbundles of maximal degree of a general V , when there are finitely many.

Keywords

Acknowledgement

The first named author was supported by the National Research Foundation of Korea: NRF-2020R1F1A1A01068699.

References

  1. V. Balaji and C. S. Seshadri, Semistable principal bundles. I. Characteristic zero, J. Algebra 258 (2002), no. 1, 321-347. https://doi.org/10.1016/S0021-8693(02)00502-1 
  2. A. Beauville, Orthogonal bundles on curves and theta functions, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 5, 1405-1418.  https://doi.org/10.5802/aif.2216
  3. A. Beauville, Y. Laszlo, and C. Sorger, The Picard group of the moduli of G-bundles on a curve, Compositio Math. 112 (1998), no. 2, 183-216. https://doi.org/10.1023/A:1000477122220 
  4. I. Biswas and T. L. Gomez, Hecke transformation for orthogonal bundles and stability of Picard bundles, Comm. Anal. Geom. 18 (2010), no. 5, 857-890. https://doi.org/10.4310/CAG.2010.v18.n5.a1 
  5. D. Cheong, I. Choe, and G. H. Hitching, Counting maximal Lagrangian subbundles over an algebraic curve, J. Geom. Phys. 167 (2021), Paper No. 104288, 20 pp. https://doi.org/10.1016/j.geomphys.2021.104288 
  6. D. Cheong, I. Choe, and G. H. Hitching, Isotropic Quot schemes of orthogonal bundles over a curve, Internat. J. Math. 32 (2021), no. 8, Paper No. 2150047, 36 pp. https://doi.org/10.1142/S0129167X21500476 
  7. I. Choe and G. H. Hitching, A stratification on the moduli spaces of symplectic and orthogonal bundles over a curve, Internat. J. Math. 25 (2014), no. 5, 1450047, 27 pp. https://doi.org/10.1142/S0129167X14500475 
  8. I. Choe and G. H. Hitching, Maximal isotropic subbundles of orthogonal bundles of odd rank over a curve, Internat. J. Math. 26 (2015), no. 13, 1550106, 23 pp. https://doi.org/10.1142/S0129167X15501062 
  9. J.-M. Drezet and M. S. Narasimhan, Groupe de Picard des varietes de modules de fibres semi-stables sur les courbes algebriques, Invent. Math. 97 (1989), no. 1, 53-94. https://doi.org/10.1007/BF01850655 
  10. W. Fulton and J. Harris, Representation theory, Graduate Texts in Mathematics, 129, Springer, New York, 1991. https://doi.org/10.1007/978-1-4612-0979-9 
  11. T. Graber, J. Harris, and J. Starr, Families of rationally connected varieties, J. Amer. Math. Soc. 16 (2003), no. 1, 57-67. https://doi.org/10.1090/S0894-0347-02-00402-2 
  12. P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, USA, 1994. 
  13. A. Grothendieck, Sur la classification des fibres holomorphes sur la sphere de Riemann, Amer. J. Math. 79 (1957), 121-138. https://doi.org/10.2307/2372388 
  14. A. Grothendieck, A General Theory of Fibre Spaces with Structure Sheaf, second ed. NSF report, Univ. of Kansas, 1958. 
  15. R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer, New York, 1977. 
  16. A. Hirschowitz, Problemes de Brill-Noether en rang superieur, C. R. Acad. Sci. Paris Ser. I Math. 307 (1988), no. 4, 153-156. 
  17. G. H. Hitching, Subbundles of symplectic and orthogonal vector bundles over curves, Math. Nachr. 280 (2007), no. 13-14, 1510-1517. https://doi.org/10.1002/mana.200510561 
  18. Y. I. Holla, Counting maximal subbundles via Gromov-Witten invariants, Math. Ann. 328 (2004), no. 1-2, 121-133. https://doi.org/10.1007/s00208-003-0475-0 
  19. H. Lange and P. E. Newstead, Maximal subbundles and Gromov-Witten invariants, in A tribute to C. S. Seshadri (Chennai, 2002), 310-322, Trends Math, Birkhauser, Basel, 2003. 
  20. A. Lanteri and R. Mallavibarrena, Projective bundles enveloping rational conic fibrations and osculation, J. Pure Appl. Algebra 224 (2020), no. 12, 106429, 21 pp. https://doi.org/10.1016/j.jpaa.2020.106429 
  21. A. Lanteri, R. Mallavibarrena, and R. Piene, Inflectional loci of quadric fibrations, J. Algebra 441 (2015), 363-397. https://doi.org/10.1016/j.jalgebra.2015.06.023 
  22. D. B. Mumford, Theta characteristics of an algebraic curve, Ann. Sci. Ecole Norm. Sup. (4) 4 (1971), 181-192.  https://doi.org/10.24033/asens.1209
  23. W. M. Oxbury, Varieties of maximal line subbundles, Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 1, 9-18. https://doi.org/10.1017/S0305004199004302