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5

In general this isn't possible. The material that follows is a bit technical, but I do not have the time to explain all the details here now. Suppose that $a, b \in T$ and that $a, b$ become squares in the completion of $T$ (see example below). Then the projective scheme $S$ defined by $a X^2 + b Y^2 + Z^2 = 0$ in $\mathbf{P}^2_T$ is $2$-dimensional with ...

8

Etale topology, required to define etale cohomology, is not a topology in the usual sense. It is Grothendieck topology only. In the category of topological manifolds, an etale cover of $X$ is a surjective local homeomorphism $Y\rightarrow X$. In the etale topology on varieties, this is "morally true" as well with local homeomorphism, replaced by a ...

1

A chapter of the unpublished PhD thesis of Rebecca Leng, a student of Miles Reid from about 2002, presents a careful study of a natural affine cover of the minimal resolution $Y={\rm GHilb}({\mathbb A^2})$ of the singularity $X={\mathbb A}^2/G$ of type D. The conclusion in Section 2.5.6 is that 􏰳􏰍she "constructs an affine open cover of \${\rm GHilb}({\...

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