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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.

5 votes

Double points in the Grothendieck ring

There is however a theory which encodes richer data, though it (at least a priori) depends on more input than just the scheme structure of $X$, and is only defined for certain kinds of schemes. …
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1 vote

Locally affine varieties and du Val singularities

A chapter of the unpublished PhD thesis of Rebecca Leng, a student of Miles Reid from about 2002, presents a careful study of a natural affine cover of the minimal resolution $Y={\rm GHilb}({\mathbb A …
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