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The first purpose of schemes theory is the geometrical study of solutions of algebraic systems of equations, not only over the real/complex numbers, but also over integer numbers (and more generally over any commutative ring with 1). It was finalized by Alexandre Grothendieck, during the 1950s and the 1960s.

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How much of scheme theory can you visualize?

I've found this set of notes http://math.mit.edu/~poonen/papers/Qpoints.pdf by Bjorn Poonen very helpful in learning arithmetic geometry. I'd read everything very carefully and do all the exercises th …
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