On the Picard Bundle

Fix a holomorphic line bundle ξ over a compact connected Riemann surface X of genus g, with g ≥ 2, and also fix an integer r such that degree(ξ) > r(2g −1). Let Mξ(r) denote the moduli space of stable vector bundles over X of rank r and determinant ξ. The Fourier–Mukai transform, with respect to a Poincar´e line bundle on X × J(X), of any F ∈ Mξ(r) is a stable vector bundle on J(X). This gives an embedding of Mξ(r) in a moduli space associated to J(X). If g = 2, then Mξ(r) becomes a Lagrangian subvariety.