 Degree-zero Picard group of a curve

For a smooth projective curve, the degree-zero Picard group consists of line bundles of degree zero up to isomorphism, with tensor product as addition. A nonzero rational section of a line bundle identifies each line bundle with $\mathcal{O}(D)$ for a divisor on an algebraic curve; degree is the sum of multiplicities weighted by residue-field degrees. Over an algebraically closed field, every degree-zero divisor is a sum of differences of points.