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Suppose $X$ is a nice (finite type over $\mathbb{C}$, smooth and proper if necessary) variety. Suppose we're given the $\mathbb{C}$-points of $X$ as a set and for any finite subset $S\subset X$ we know the DG algebra of derived sections $\Gamma_{DG}(O_X, X\setminus S)$ of the sheaf of functions, together with the transition maps when we replace $S$ by $S'\supset S$. Can we reconstruct $X$?

Edit: I think the answer to this question is "no", and here is a better version of the question: How much of a variety can be reconstructed from codimension-zero data?

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It's not immediately clear (although perhaps it is possible to make clear) how you can "know" all finite subsets of $X$ without knowing $X$. More details in the q maybe? – user30035 Mar 24 '13 at 11:36
PS if you take any smooth surface then removing finitely many points won't change the sections (because zeros of sections show up in codim 1). I don't know what DG means though so I don't even know if this observation is of relevance. – user30035 Mar 24 '13 at 11:38

I realized the answer is almost certainly "no", so I asked a better version of this question at How much of a variety can be reconstructed from codimension-zero data?.

I think that if you take two hypersurfaces in $\mathbb{P}^3$ of the same degree defined by two polynomials with mutually transcendental coefficients then they won't be distinguishable. The cohomology of the constant sheaf for a surface with punctures is concentrated in degrees $0,1$ and has only $\mathbb{C}$ in degree $0$, the DG algebra is uniquely determined by the vector space $H^1(O_X,X\setminus S)$ and therefore should boil down to some combinatorial data which generically won't depend on coefficients. (I'm not 100% sure of this since there may be some deformable data in the transition maps as you add points to $S$).

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