I have a precise reference question: can you recommend me a book (or more probably a chapter of a book, or a paper) that explains and motivates the transition from the classical (more geometrical) point of view on the algebraic geometry to Grothendieck's point of view? In my course we started with some results about functors and representability, in order to introduce the affine spaces as functors on algebras, state the Hilbert's Nullstellensatz and see some basic results of classical algebraic geometry. Then, we defined the geometric points, and began to study the spectrum of rings as a topological spaces. I would like a reference that explains well the sense of this last "generalization", and that gives some motivation, because the notes of my course are very dry.
I don't know if this question is off topic or not; it could be subjective, but I don't think that there are so much books to choose from, so it shouldn't be too subjective. Thank you in advance
If you haven’t seen it before, Ravi Vakil’s book is full of fantastic explanations, motivations and examples http://math.stanford.edu/~vakil/216blog/FOAGnov1817public.pdf