For more than 10 years before the apearance of the Grothendieck's theory on schemes, the Weil's foundations of algebraic geometry had been the standard language of algebraic geometry. There were important works written in the language. So a natural question is: How can they be translated in the language of schemes? There is no general solution to this problem and one has to prove each one of them from scratch?

For example, Serre wrote in his book "Algebraic groups and class fields":

This chapter contains the construction and elementary study of the generalized Jacobians of an algebraic curve. We will follow closely the paper of Rosenlicht [64] on this subject, itself inspired by Weil's Varietes abeliennes [89], where the case of the usual Jacobian is treated. We will make use, as they did, of the method of "generic points". This obliges us to renounce the point of view of the preceding chapters (where all points had their coordinates in a fixed base field), and to adopt that of Foundations [87]. It is certain that the generic points could be replaced by divisors on product varieties, after first developing in detail the properties of these divisors (that is to say essentially the cohomology of coherent algebraic sheaves on a product variety); that would take us too far afield.

How the generic points could be replaced by divisors on product varieties?