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!


In my humble opinion I would say that it's a good idea to know about smooth manifolds before studying algebraic geometry, but not 100% necessary. The language of algebraic geometry is different enough from the language of smooth manifolds that in any book on algebraic geometry (Hartshorne, for example), every new concept is defined and I'm pretty sure that no background in smooth manifolds is assumed. That being said, however, many ideas in algebraic geometry come from manifold theory, and if you haven't seen the ideas before they may seem strange. For example, the intuition for sheaves, vector bundles, tangent space, etc., comes from manifold theory, and if you haven't worked with vector bundles before, for instance, it may seem strange at first in the context of algebraic geometry. Now if you already have a notion of these objects, I don't see why there should be any problem.


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