I am a student learning algebraic geometry. Recently I came across the book "Principles of Algebraic Geometry" by Griffiths and Harris and I noticed that they do not ever talk about schemes.
I am sure that many students spend a long time trying to learn about schemes. For example, one might learn be first introduced to varieties and then beginning reading about schemes from Hartshorne's book. It seems to me, the novice, that although the language of schemes is without a doubt important, there are many situations in algebraic geometry where one can get away without thinking about them at all.
For example, last semester I spent time working through the technical details about schemes using references such as Vakil's notes and Hartshorne. This semester, I spent time learning about birational geometry, and I couldn't help but feel that many of the things about schemes I spent time learning were not particularly useful.
This is not a post asking why schemes are interesting or why students should learn about schemes, there are plenty of reasons in the posts below.
What should be learned in a first serious schemes course?
What elementary problems can you solve with schemes?
Modern algebraic geometry vs. classical algebraic geometry
I'm hoping some algebraic geometers can weigh in on the following: What role does the language of schemes play in modern algebraic geometry? It seems that some algebraic geometers, e.g. the birational geometers or Hodge theorists, don't work so closely with schemes as much as others, e.g. moduli space people.
Thank you very much!
