History of Algebraic Geometry: Morphisms and Birational Geometry

Question

Good people, I'm trying to get my head around the history of algebraic geometry, and while Dieudonné's tome is a very good source (very often the only source), it can from time to time be very idiosyncratic. As such, I was wondering if I had understood things properly on a certain point. Specifically, is the following statement correct:

• The notion of morphisms between varieties as we understand it today with regular maps and all that is an invention of Zariski and Weil in the 30s and 40s. Prior to that, the only notion of "morphisms" that algebraic geometers had were rational transformations (i.e. rational maps), and the only notion of two varieties being "isomorphic" that they had was that two varieties were birationally equivalent.

If that is true, then would it further be fair to say the following:

• The very notion of thinking of varieties and morphisms between them being the central "things" around which everything in algebraic geometry centers is a conception that first came about due to category theory, and that prior to Zariski setting out on his project to bring algebraic geometry on more algebraic footing, people didn't have that particular structured approach to algebraic geometry. That in a sense, it was more akin to "trying to do Euclid and Descartes on algebraic surfaces". Indeed, prior to Weil's 1946 book Foundations of Algebraic Geometry, I have so far been unable to find any text talking about morphisms between varieties, either that uses the word morphism or that describes the kind of mapping between two varieties that we would call a morphism by today's terminology.

As always, look forward to your responses.

Answer 1

Surely the claim that pre-Grothendieck, algebraic geometry only had rational maps, cannot be correct: The study of algebraic surfaces, which I think led to the consideration of birational maps as a means of having a tractable classification problem involves considering embedded curves and blow-ups and blow-downs in a way that makes it extremely unlikely that algebraic geometers of that period did not have a notion of a morphism of varieties. (Their terminology may have been different of course).

On the other hand, the extent to which the notion of a scheme received much attention before Grothendieck is I think limited -- I think the term is due to Grothendieck, but I don't know the extent to which they had been studied in other guises -- if at all -- before him. The bold idea that objects like $\text{Spec}(\mathbb Z)$ should be viewed as geometric spaces is I think due to Grothendieck's school.

Update: I am sure there are better references, but this paper by C. Segre in 1904 considers "algebraic transformations" and "correspondences" which appear to include birational transformations, but be more general -- he mentions projective, birational and conformal transformations, and makes a point of saying it is easy to imagine new kinds of "geometric transformations" that one might wish to study. Moreover he has a notion of an "algebraic manifold" which seems to be reasonably close to that of an algebraic variety

I fear it is not quite the reference you need, but from skim-reading it, my impression is that he would not have found anything particularly new or surprising if someone had presented him with the definition of varieties and morphisms between them. (Again, schemes might be a different story!)