Completion, divisors and degree on smooth curves

PDF p. 13: Divisibility of the normalization Picard group

For a regular integral separated curve of finite type over a perfect field, there is a smooth projective integral curve containing it as a dense open subscheme with finite closed-point complement; if the original curve is affine, that complement is nonempty. On a smooth integral finite-type curve, the divisor class group, defined as finite integer sums of closed points modulo principal divisors of nonzero rational functions, identifies naturally with the Picard group by sending a closed point to its associated line bundle. Restriction to a nonempty open subset corresponds to deleting divisor coefficients outside that subset. On a smooth projective curve over an algebraically closed field, degree is an additive homomorphism on the Picard group, the class of a closed point has degree equal to its residue-field degree, and its kernel is the group of rational points of the degree-zero Picard variety.

The completion and nonproperness assertions are imported from the first reference. Its curve convention includes separatedness (Definitions 33.43.1 and 33.3.1); the completion assertion uses that convention explicitly. The subsequent divisor comparison retains its locally Noetherian integral scope. For the divisor comparison, smooth-curve local rings are fields or discrete valuation rings, hence UFDs; the second reference supplies the isomorphism. Its construction by orders of meromorphic sections is compatible with restriction because the same local rings and orders occur at points of the open subset. The third reference gives the degree formula, using that a closed point on a smooth curve is an effective Cartier divisor. The fourth gives the stated Picard-variety identification. These are external inputs; their proofs are not claimed to have been reconstructed here.