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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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Research in applied algebraic geometry that essentially needs a background of modern algebra...
potentially applicable", and the issue of whether something could be translated into classical language, and simply offer an example that I came across recently:
Max Lieblich, Lucas Van Meter: Two Hilbert Schemes … As the title suggests, this is a paper on the geometry of computer vision that uses Hilbert schemes, not to mention Artin stacks.( But really it's not as fancy as all that: in the end they turn out just …