  There is a classical relationship in algebraic geometry between a
hyperelliptic curve and an associated pencil of quadric hypersurfaces. We
investigate symplectic aspects of this relationship, with a view to
applications in low-dimensional topology. We construct a derived equivalence
between the Fukaya category of a curve and the nilpotent summand of the Fukaya
category of the associated complete intersection of two quadrics. This
essentially determines the instanton Floer homology of a 3-manifold fibred by
genus two curves.
