Prerequisite of Algebraic Geometry Algebraic geometry, as far as I know, is a very important branch of mathematics, which is also very difficult. I am going to take a try to taste that. Before really going into the field, I have two questions that could be a guide for me.


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*Is it really necessary to know about algebraic geometry, even for a professional mathematician focus on other fields?

*I've learned from Wikipedia that commutative algebra is subsumed into algebraic geometry. So do I need to study commutative algebra or projective geometry before starting learning algebraic geometry?
 A: For your first question, it really depends.  If you're going into a sub-branch of algebra, you will very likely have at least a little interaction with algebraic geometry.  Knowing some of the basic ideas and terminology is useful, but if you were going to need much more than that, you would know it well in advance.  If you are not going into algebra, but you were going into something involving geometry, you may end up doing some things involving algebraic geometry, but it is significantly less likely.  If you go into analysis or logic, it is very unlikely (but not impossible) for you to come across thing involving algebraic geometry.
For your second question, modern algebraic geometry is definitely built on commutative algebra, and you can't play around with quasi-coherent sheaves over schemes if you don't have a solid footing in commutative algebra.  However, there is a compelling argument to be made that one should learn classical algebraic geometry and some differential geometry (at least to the point of fiber bundles) before tackling any of the modern stuff, as the modern approach can obscure the geometric ideas behind an algebraic edifice, leaving you without intuition or motivation for many ideas and definitions.
The prerequisites for studying classical algebraic geometry are significantly more humble, and the commutative algebra needed could easily be learned as you go along.  In fact, that is probably a good idea, as many constructions in commutative algebra are motivated by geometric concerns, meaning that concurrent study enriches both subjects.
