The intrinsic Lagrangian fibration on the cotangent bundle of an intersection of two quadrics coincides with the Hitchin morphism of the moduli space of twisted Spin bundles over the associated hyperelliptic curve.
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Intersection of two quadrics: modular interpretation and Hitchin morphism
The intrinsic Lagrangian fibration on the cotangent bundle of an intersection of two quadrics coincides with the Hitchin morphism of the moduli space of twisted Spin bundles over the associated hyperelliptic curve.