In what sense does a divisor actually divide?

So I've been stuck for the past couple of days trying to figure out how a divisor in algebraic geometry is actually, in some sense, analoguous to a divisor in elementary number theory. I cannot help but feel that if I understood this, I would have an easier time understanding the "point" of the concept.

Several texts give hints at the connection. In Eisenbud's wonderful Commutative Algebra: with a view toward Algebraic Geometry the Berkeley professor writes (p. 262-3):

If $a,b$ are elements of a ring, then $b$ divides $a$ iff $a \in (b)$. In general, an ideal can be regarded as something by which an element might be divisible--a "divisor." Because of the unique factorization into prime ideals in a Dedekind domain, nonzero ideals there correspond to finite sets of codimensions 1 prime ideals, each with multiplicity. The term divisor was transferred to such sets and stuck there.

Which of course brings to mind Kummer's original idea when he first defined ideal numbers which Dedekind eventually repurposed to our modern notion of an ideal. Nevertheless, I am still far from grasping the connection fully.

Specifically, if $V$ is some variety and $C$ is a Weil divisor of $V$, then how does $C$ "divide" $V$?

As Jake Levinson mentioned in comment, you should think of $C$ as a divisor on $V$ rather than divisor of V.

Maybe this example on $V = \operatorname{Spec}\mathbb{Z}$ would help.

• Irreducible divisors in this case are the (usual) primes $(p)$.
• A Weil divisor $C$ is then a formal sum $C = \sum e_p (p)$, where $e_p \in \mathbb{Z}$.
• A positive integer $n$ is then a global section/function on $V$. If the prime factorization of $n = \prod_p p^{e_p}$, then the principal divisor $div(n)$ is exactly $\sum e_p (p)$. This may more closely match with your intuition of "divisors".

Since you mentioned ideal numbers, you can play the same game with Spec of integers in a number field. The irreducible divisors is in that case correspond to prime ideals of your integers, and principal divisors again correspond to a factorization of principal ideal into product of primes.