You may find the paper [Generalized divisors on Gorenstein curves and a theorem of Noether](https://doi.org/10.1215/kjm/1250520873) of Hartshorne to be helpful.


  [1]: https://projecteuclid.org/journals/kyoto-journal-of-mathematics/volume-26/issue-3/Generalized-divisors-on-Gorenstein-curves-and-a-theorem-of-Noether/10.1215/kjm/1250520873.full