Degree of a line bundle ($\mathrm{deg}L=\mathrm{deg}D\text{when }L\cong {\mathcal{O}}_C(D)$)

On a smooth projective curve, a nonzero rational section represents a line bundle by its divisor on an algebraic curve. The degree is the sum of its vanishing orders minus its pole orders, including residue-field degrees over a nonclosed field. Principal divisors on an algebraic curve have degree zero, so this number is independent of the section. A nonzero regular section gives an effective divisor and therefore a nonnegative degree. Degrees add under tensor products and change sign under duality.