I am searching for a scheme $X$ which can be obtained by gluing two affine schemes (along open subsets) such that: 1) X is non-reduced; 2) $\Gamma(X,\mathcal O_X)$ is a reduced ring. Any ideas? Thanks in advance.
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Hint:
Can you construct a ''fat'' projective line of some sort? Namely, remember that $\text{Spec } k[x]$ is the affine line and we get projective space by gluing together two affine lines. Can you do something similar by fattening up the lines you glue?
My example was the following. Take $\text{Proj } k[x,y,z]/z^2$. I believe that working on affines, this should produce something with only $k$ as global sections. I haven't worked it out in detail however.
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$\begingroup$ What is your specific example? A fat projective line as a fat ring of global sections, basically because you cannot glue along the fat variable. $\endgroup$ Oct 24, 2014 at 14:43
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$\begingroup$ I am also thinking in this direction, but I find it difficult to prevent the nilpotent sections to be glued together and form a nilpotent global section. $\endgroup$– AYKOct 24, 2014 at 14:45
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$\begingroup$ @MartinBrandenburg See my example here. I hope it works, I haven't worked it through in detail as I wrote. $\endgroup$ Oct 24, 2014 at 14:50
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1$\begingroup$ $\mathrm{Proj}(k[x,y,z]/(z^2)) = D_+(x) \cup D_+(y)$ (since $D_+(z)=\emptyset$) with $D_+(x) = \mathrm{Spec}(k[y/x,z/x]/(z/x)^2)$ and $D_+(y) = \mathrm{Spec}(k[x/y,z/y]/(z/y)^2)$. I don't think that the ring of global sections contains $z$. It has a good chance to be $k$. $\endgroup$ Oct 24, 2014 at 15:05
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$\begingroup$ I have computed this and barring mistakes, I believe that this has global sections $k$. $\endgroup$ Oct 24, 2014 at 15:06