# 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-Weil-Grothendieck, 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".

As always, look forward to your responses.

• Dieudonné's published a short version of his tome, which is somewhat easier to follow. While the first passage is close to right, I always thought that Weil and Zariski's work was conceptually set-theoretic, not categorical. Weil, in particular, was adapting Elie Cartan's differential-geometric methods, he was also behind Bourbaki's set-theoretic "structures". Oct 15 at 20:46
• It was Grothendieck, somewhat anticipated by Chevalley and Serre, that effected the second transformation in 1950-s, with schemes, morphisms and other categorical language and methods. Dieudonne even separates his "7th epoch" from their "6th". Cartier has a good essay focusing on his innovations, see also McLarty's paper, and his answer in the Grothendieck thread on HSM. Oct 15 at 20:57