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Lagrangian subbundles of symplectic bundles over a curve

Published online by Cambridge University Press:  22 February 2012

INSONG CHOE
Affiliation:
Department of Mathematics, Konkuk University, 1 Hwayang-dong, Gwangjin-Gu, Seoul 143-701, Korea. e-mail: [email protected]
GEORGE H. HITCHING
Affiliation:
Høgskolen i Oslo og Akershus, Postboks 4, St. Olavs plass, 0130 Oslo, Norway. e-mail: [email protected]

Abstract

A symplectic bundle over an algebraic curve has a natural invariant sLag determined by the maximal degree of its Lagrangian subbundles. This can be viewed as a generalization of the classical Segre invariants of a vector bundle. We give a sharp upper bound on sLag which is analogous to the Hirschowitz bound on the classical Segre invariants. Furthermore, we study the stratifications induced by sLag on moduli spaces of symplectic bundles, and get a full picture for the case of rank four.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2012

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References

REFERENCES

[1]Adkins, W. and Weintraub, S.Algebra: an approach via Module theory. Graduate Texts in Mathematics 136 (Springer-Verlag, 1992).CrossRefGoogle Scholar
[2]Biswas, I. and Gomez, T.Hecke correspondence for symplectic bundles with application to the Picard bundles. Internat. J. Math. 17, no. 1 (2006), 4563.CrossRefGoogle Scholar
[3]Brambila–Paz, L. and Lange, H.A stratification of the moduli space of vector bundles on curves. J. Reine Angew. Math. 494 (1998), 173187.CrossRefGoogle Scholar
[4]Choe, I. and Hitching, G. H.Secant varieties and Hirschowitz bound on vector bundles over a curve. Manuscripta Math. 133 (2010), 465477.CrossRefGoogle Scholar
[5]Hartshorne, R.Algebraic geometry. (Graduate Texts in Mathematics 52) (Springer-Verlag, 1977).CrossRefGoogle Scholar
[6]Hirschowitz, A.Problèmes de Brill–Noether en rang supérieur. Prepublications Mathématiques n. 91, Nice (1986).Google Scholar
[7]Hitching, G. H. Moduli of symplectic bundles over curves. Doctoral dissertation (University of Durham, 2005).Google Scholar
[8]Hitching, G. H.Subbundles of symplectic and orthogonal vector bundles over curves. Math. Nachr. 280, no. 13–14 (2007), 15101517.CrossRefGoogle Scholar
[9]Hitching, G. H.Moduli of rank 4 symplectic vector bundles over a curve of genus 2. J. London Math. Soc. (2) 75, no. 1 (2007), 255272.CrossRefGoogle Scholar
[10]Holla, Y. I. and Narasimhan, M. S.A generalisation of Nagata's theorem on ruled surfaces. Comp. Math. 127 (2001), 321332.CrossRefGoogle Scholar
[11]Hwang, J.-M. and Ramanan, S.Hecke curves and Hitchin discriminant. Ann. Sci. École Norm. Sup. (4) 37, no. 5 (2004), 801817.CrossRefGoogle Scholar
[12]Kempf, G. R.Abelian integrals. Monografías del Instituto de Matemáticas 13 (Universidad Nacional Autónoma de México, 1983).Google Scholar
[13]Kempf, G. R. and Schreyer, F.-O.A Torelli theorem for osculating cones to the theta divisor. Compositio Math. 67, no. 3 (1988), 343353.Google Scholar
[14]Lange, H.Universal families of extensions. J. Algebra 83, no. 1 (1983), 101112.CrossRefGoogle Scholar
[15]Lange, H. and Narasimhan, M. S.Maximal subbundles of rank two vector bundles on curves. Math. Ann. 266, no. 1 (1983), 5572.CrossRefGoogle Scholar
[16]Mukai, S. and Sakai, F.Maximal subbundles of vector bundles on a curve. Manuscripta Math. 52 (1985), 251256.CrossRefGoogle Scholar
[17]Nagata, M.On Self-intersection number of a section on a ruled surface. Nagoya Math. J. 37 (1970), 191196.CrossRefGoogle Scholar
[18]Narasimhan, M. S. and Ramanan, S.Deformations of the moduli space of vector bundles over an algebraic curve. Ann. Math. (2) 101 (1975), 391417.CrossRefGoogle Scholar
[19]Ramanan, S. Orthogonal and spin bundles over hyperelliptic curves. Geometry and Analysis: Papers dedicated to the memory of V. K. Patodi (Springer–Verlag, 1981).CrossRefGoogle Scholar
[20]Ramanathan, A.Moduli for principal bundles over algebraic curves I & II. Proc. Indian Acad. Sci. Math. Sci. 106, no. 3 (1996), 301–328 and no. 4 (1996), 421449.CrossRefGoogle Scholar
[21]Russo, B. and Teixidor i Bigas, M.On a conjecture of Lange J. Algebraic Geom. 8 (1999), 483496.Google Scholar
[22]Serman, O.Moduli spaces of orthogonal and symplectic bundles over an algebraic curve. Compositio Math. 144, no. 3 (2008), 721733.CrossRefGoogle Scholar
[23]Terracini, A.Sulle Vk per cui la varietà degli Sh (h + 1)-seganti ha dimensione minore dell'ordinario. Rend. Circ. Mat. Palermo 31 (1911), 392396.CrossRefGoogle Scholar