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Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

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Lifting sections to completion of closed subschemes

No. Take $R=k[x, y]$ and $I = (x)$ and $J = (y)$. Then let $V$ be the glueing of two copies of $\text{Spec}(R)$ at $0 = (0, 0)$. Let $s_{IJ}$ be the morphism which sends $V(I)$ into the first copy and …
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