Moduli of rank 4 symplectic vector bundles over a curve of genus 2

Hitching, George H.

doi:10.1112/jlms/jdl011

Cited by 8 publications

(12 citation statements)

References 12 publications

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“…In [5], these techniques are used in the construction of a generically finite cover of M X (Sp 2 C) when X has genus 2. This explicit description is currently being applied to the study of theta divisors of bundles in M X (Sp 2 C).…”

Section: Covers Of Moduli Spacesmentioning

confidence: 99%

Subbundles of symplectic and orthogonal vector bundles over curves

Hitching¹

2007

Mathematische Nachrichten

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We review the notions of symplectic and orthogonal vector bundles over curves, and the connection between principal parts and extensions of vector bundles. We give a criterion for a certain extension of rank 2n to be symplectic or orthogonal. We then describe almost all of its rank n vector subbundles using graphs of sheaf homomorphisms, and give criteria for the isotropy of these subbundles. Finally, we sketch the use of these ideas in moduli questions for symplectic vector bundles.

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Section: Covers Of Moduli Spacesmentioning

confidence: 99%

Subbundles of symplectic and orthogonal vector bundles over curves

Hitching¹

2007

Mathematische Nachrichten

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“…In particular when n = 1, since Sp 1 is isomorphic to SL 2 , this reduces to the stratification already studied on the moduli space SU(2, O X ) of semistable bundles of rank two with trivial determinant. We prove the following result on the stratification on the moduli space In the case of genus two, this was proven in [9]. The key ingredient of the proof there was a symplectic version of Lange and Narasimhan's description [15] of the Segre invariant using secant varieties (also see [4] for a higher rank version in the case of vector bundles).…”

Section: Theorem 1•4 For Every Symplectic Bundle W Of Rank 2n We Havementioning

confidence: 93%

“…The key ingredient of the proof there was a symplectic version of Lange and Narasimhan's description [15] of the Segre invariant using secant varieties (also see [4] for a higher rank version in the case of vector bundles). In this paper, we generalize the method and results of [9] to the case of arbitrary genus. We will see an interesting variant of Lange and Narasimhan's picture in the case of symplectic bundles: a relation between the invariant s Lag and the higher secant varieties of certain fibre bundles over X .…”

Section: Theorem 1•4 For Every Symplectic Bundle W Of Rank 2n We Havementioning

confidence: 99%

Lagrangian subbundles of symplectic bundles over a curve

Choe¹,

Hitching²

2012

Math. Proc. Camb. Phil. Soc.

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AbstractA symplectic bundle over an algebraic curve has a natural invariantsLagdetermined 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 onsLagwhich is analogous to the Hirschowitz bound on the classical Segre invariants. Furthermore, we study the stratifications induced bysLagon moduli spaces of symplectic bundles, and get a full picture for the case of rank four.

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“…An important ingredient is a description of this M 2 from [13] using vector bundle extensions. Let W → X be a symplectic or orthogonal bundle of rank 2n and E ⊂ W an Lagrangian subbundle (that is, isotropic of maximal rank n).…”

Section: Generalised Theta Divisors and The Theta Mapmentioning

confidence: 99%

Rank Four Symplectic Bundles Without Theta Divisors Over a Curve of Genus Two

Hitching

2008

Int. J. Math.

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The moduli space M 2 of rank four semistable symplectic vector bundles over a curve X of genus two is an irreducible projective variety of dimension ten. Its Picard group is generated by the determinantal line bundle Ξ. The base locus of the linear system |Ξ| consists of precisely those bundles without theta divisors, that is, admitting nonzero maps from every line bundle of degree −1 over X. We show that this base locus consists of six distinct points, which are in canonical bijection with the Weierstrass points of the curve. We relate our construction of these bundles to another of Raynaud and Beauville using Fourier-Mukai transforms. As an application, we prove that the map sending a symplectic vector bundle to its theta divisor is a surjective map from M 2 to the space of even 4Θ divisors on the Jacobian variety of the curve.

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Moduli of rank 4 symplectic vector bundles over a curve of genus 2

Cited by 8 publications

References 12 publications

Subbundles of symplectic and orthogonal vector bundles over curves

Subbundles of symplectic and orthogonal vector bundles over curves

Lagrangian subbundles of symplectic bundles over a curve

Rank Four Symplectic Bundles Without Theta Divisors Over a Curve of Genus Two

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