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Feb 6, 2012 at 10:46 vote accept MZWang
Aug 29, 2011 at 19:40 comment added Georges Elencwajg No gluing is necessary. Consider $\mathbb P^2_k$ (with coordinates (u:v:w)). Kevin's scheme is the locally closed subscheme obtained by first taking the cross $uv=0$ (a closed subscheme of $\mathbb P^2_k$) and then deleting the closed point $(0:1:0)$ (say) from that cross.
Aug 29, 2011 at 18:09 comment added Kevin Ventullo @unknown: What Wilberd said is correct. The subvariety $X\cup P/\lbrace y\rbrace$ will be affine with global sections isomorphic to $k[S,T]/\langle ST\rangle$. In general, it is possible to make sense of gluing along closed subschemes. See this question: mathoverflow.net/questions/64294/gluing-along-closed-subschemes.
Aug 29, 2011 at 13:37 comment added Wilberd van der Kallen @unknown I do not know. But the attaching along the point $x$ in the example is clear: Locally around the point $x$ it looks like the union of two axes in the affine plane. And away from the point it just looks like it looked before the attaching. If you wish you can look at the result as a push out, say as the coequalizer of a map $x\to \mathbb P^1$ and a map $x\to \mathbb A^1$. At the ring level this is locally given as a fibered product.
Aug 29, 2011 at 12:34 comment added MZWang Thanks! But excuse me, what is attach along a closed subscheme?
Aug 29, 2011 at 8:11 comment added Georges Elencwajg Very nice, Kevin!
Aug 29, 2011 at 6:42 history answered Kevin Ventullo CC BY-SA 3.0