I have a chance to take a course in algebraic geometry developed in the scheme theoretic setting and am considering my options.
I have taken graduate courses in general and commutative algebra and know the very basics of classical algebraic geometry.
I have, however, not taken a course in smooth manifolds. Now I can in fact take a course in smooth manifolds this semester as well, but for a number of reasons I'd prefer to wait. One of these reasons is that I have plenty of other coursework and I am not sure I will be able to properly focus if I take this course as well.
My question is then aimed at people who know modern algebraic geometry:
Is it completely crazy to study ''hard core'' algebraic geometry without proper background in smooth geometry?
(I am not oblivious to the notion of smooth manifolds, for instance I do understand that the smooth functions on a manifold form a sheaf of rings.)
Thank you very much for your thoughts!