 Saturated line subbundle on a smooth curve

A rank-one subsheaf of a vector bundle on a smooth algebraic curve is saturated when the quotient is torsion-free. Intersecting a one-dimensional subspace of the generic fibre with each stalk constructs a saturated subsheaf. The local ring of a smooth algebraic curve is a discrete valuation ring, so the subsheaf and its torsion-free quotient are locally free. Saturating a nonzero map from a line bundle enlarges that line bundle by an effective divisor, recording the map's zeros.