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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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Minimal fields of isomorphism for varieties

This question is most "reasonable" when $V$ is projective, since at least then the automorphism functor is represented by a (locally finite type) $K$-scheme $G_V := {\rm{Aut}}_{V/K}$ and so one can tr …
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