I would like to better understand a certain compactification of the Jacobian variety of a singular algebraic plane curve as described in [Cook's Ph.D. thesis][1]. So far, I have only been able to find references discussing this compactification for the case of simple/ADE singularities (e.g.[these][2] [papers][3]). 

**Question:** Is there some kind of obstruction and/or limitation of our current tools that prevents this compactification from working for arbitrary singularities? What parts of the aforementioned construction in Cook's thesis crucially rely on the assumption that every singularity of the curve is simple?

  [1]: https://livrepository.liverpool.ac.uk/3175416/1/357430.pdf
  [2]: https://arxiv.org/abs/2405.11860v1
  [3]: https://arxiv.org/abs/1012.5541