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Picard group is a recurring notion that I've been seeing very often recently recently. For projective varieties, Picard group is the divisor class group. For Dedekind domains, it is the ideal class group. For rings, Picard group can be defined on its algebraic line bundles. In the Wikepedia page, it says it can be defined for any ringed space. Or even more generally, it can be defined for any symmetric monoidal category. But I have yet to find a good exposition to link these together.

So my questions are:

  1. How does one see that the Picard groups in the cases of divisor class groups, ideal class groups and line bundles are related?

  2. Follow up to (1.), what are some examples (using the relation of Picard groups in different settings) that allow us to understand a number theoretic problem geometrically, or a geometry problem arithmetically?

  3. Are there results or properties of Picard groups in the general settings (ringed spaces or symmetric monoidal categories) that turn out to be significant when we examine it in the more specific settings?

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Let's correct a misconception first, before moving on to the rest of the question: it is not true that the Picard group is the divisor class group on an arbitrary projective variety. The condition that the Picard group and divisor class group coincide for a general scheme is that our scheme be factorial - see for instance here. This condition exactly tells us that Weil divisors and Cartier divisors coincide, so the divisor class group and the Picard group coincide.

In general, the Picard group is the group of isomorphism classes of invertible objects for the monoidal structure. In most cases in algebraic geometry, this means invertible sheaves under tensor product: this covers all three scenarios in your question (1). The equivalence of divisors and line bundles on factorial schemes has already been mentioned in the first paragraph. For the second paragraph, fractional ideals in a Dedekind domain $D$ are exactly the invertible sheaves on $\operatorname{Spec} D$.

As far as examples of using the Picard group, computing intersection numbers is a big one. One can do an incredible amount of geometry this way, see for instance Hartshorne chapter 5 on algebraic surfaces. Unsurprisingly, this generalizes to the whole topic of intersection theory, which you can get a lot done with. We also can say a lot with tools developed from the Picard group when studying abelian varieties. Briefly, the Jacobian variety of a curve $C$ is the connected component of the identity in the Picard group, and this concept gets a ton of use in that field. I'm sure there a wealth of examples from number theory, but I am less familiar with that area and do not have a good one to give.

As far as your final question, I'm not sure about specific insights one gets when zooming in from the very abstract picture to a more elementary picture (perhaps someone else will chime in and enlighten us). It seems to me to be more the case that invertible objects are useful in lots of situations, and the Picard group and it's generalizations are a good way to keep track of this information.

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