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I'm somewhat new to algebraic geometry. I'm currently studying algebraic curves primarily over closed fields, (for future discussion let's just call such a curve C). I was taught to think of things like $\rm{Pic}^{0}(C)$ as a set of equivalence classes of divisors on C. From what I gather, these are usually called Weil Divisors.

Some literature seems to take the approach that these classes on $\rm{Pic}^{0}(C)$ can be thought of instead as a line bundle. For example, if I let $\epsilon$ be a two-torsion point on C, I can think of $[\epsilon]$ as an order two line bundle.

My question is what the connection is. How can I intuitively see that this must be true?

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    $\begingroup$ This is the equivalence between Weil divisors and Cartier divisors. You may wish to check out Vakil's FOAG chapter 14 section 2, some of which I've summarized here. There are many other resources about this (ie here's the search results on this website), and I'd encourage you to check them out and make your question more specific. $\endgroup$
    – KReiser
    Jul 25, 2020 at 19:06
  • $\begingroup$ Thanks, I think this is exactly what I was looking for. I'll let you know if have further questions. $\endgroup$ Jul 25, 2020 at 19:51

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