Explicit Noether-Lefschetz for arbitrary threefolds

It is well-known in algebraic geometry that the geometry of a given variety is influenced by the geometry of its subvarieties. It less common, but not unusual, that a given ambient variety forces to some extent the geometry of its subvarieties. A particularly nice case of the latter is given by line bundles, whose properties do very much influence the geometry. If Y is a smooth variety and i : X →֒ Y is a smooth divisor, there is then a natural restriction map i : Pic(Y ) → Pic(X)