# History of Algebraic Geometry: Morphisms and Birational Geometry

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.

• I have the feeling that the names Kronecker, Hilbert, König, Macauley, vanderWaerden should play a role here, capturing the time span from 1880 to 1930, but perhaps that counts as pre-history? // There is a dedicated "History of science and mathematics" portal here, hsm.SE, where you might find more experts for this area of knowledge. Commented Oct 12, 2023 at 13:14
• A nice starting point is hsm.stackexchange.com/questions/1813/… Commented Oct 12, 2023 at 14:02

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!)

• I think I expressed myself poorly. By Zariski-Weil-Grothendieck, I meant the period of rapid change in algebraic geometry that occurred from the 1930s through to the 1950s, not Grothendieck in particular. Specifically, I cannot find any references to morphisms between varieties prior to Weil's Foundations of Algebraic Geometry from 1946. Even Zariski's Algebraic Surfaces from 1935 doesn't contain it. The only notion of a morphism in the sense of 'the standard kind of mapping between two varieties' that I can find prior to that are rational transformations. Commented Oct 12, 2023 at 18:00
• Perhaps the reference I have linked to in the updated answer helps a little? I'm not sure Segre would have found it important to agree on "the standard kind of transformation" between varieties, but it seems clear he envisages more than just the class of birational maps. Commented Oct 12, 2023 at 19:03
• That is a good reference! Alas, at the moment I cannot access it, but I'm sure that once I've been able to download it and have a look at it, it'll help me in establishing the sense in which these old algebraic geometers considered two varieties to be "similar". Commented Oct 12, 2023 at 19:43
• @StormyTeacup the AMS claims to makes the archive of the Bulletin freely available, so I hope you are able to access it somehow -- apologies I should have checked that before referring to it! Commented Oct 12, 2023 at 23:19
• Well perhaps it's just me not being able to figure out how that website work? :P Nevertheless, I was able to obtain that article from elsewhere now, and it looks very interesting, and has so far led me to this paper written (in Italian) by Segre called Introduzione alla geometria sopra un ente algebrico semp lice men le infinito. It's going to take me a while to translate it, but, I will say, it looks promising! :) Commented Oct 13, 2023 at 6:01