= 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.
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