TITLE: Degree of a divisor on surfaces
QUESTION [5 upvotes]: I'm reading surfaces from ALgebraic Geometry of Robin Hartshorne. On chapter $5$ page $357$ there is an affirmation: "A divisor on a surface is a sum of curves, so (in the absence of a projective embedding) it does not make sense to talk about the degree of a divisor, as in the case of curves."
Why is that? The degree would be the formal sum of the coefficients of the divisor, so those numbers will only exist if there is a projective embedding?
Thanks for any suggestion

REPLY [2 votes]: You could define the degree of a divisor as the sum of the coefficients, but that is not a useful notion —in other words, it does not have the properties we want the degree to have.
The problem is that what you really need is not on take into account the coefficients but also, somehow, the 'degree' of the components of the divisor themselves, and that is not well defined.