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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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Orbits of group scheme action
Presumably you meant to assume the schemes are finite type over $k$. … fields (e.g., one uses the robust theory of quotients modulo possibly non-smooth closed subgroup schemes). …