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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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Applications of schemes to mathematical physics

Could anyone cite some applications or developments in mathematical physics or string theory that use schemes? … However I am looking for specific (underived) scheme constructions like Fano schemes, Hilbert schemes, scheme-theoretic interesections etc …