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Given schemes $X,Y$ and $Z$ such that $Z$ is a closed subscheme of both $X$ and $Y$ the pushout exists in the category of schemes. So in particular one can glue schemes along a closed pointor glue a scheme to itself along a pair of closed points (although one needs to choose a subscheme structure - you would want to take the reduced induced subscheme structure probably). A reference for this (carried out via the category of locally ringed spaces) is given in this paper of Schwede (Corollary 3.9).

In general though the pushout in the category of locally ringed spaces need not be a scheme even if one pushes out along a subscheme - see for instance Example 3.3 in Schwede's paper.

2 Fixed last statement as in Andrew's comment

Given schemes $X,Y$ and $Z$ such that $Z$ is a closed subscheme of both $X$ and $Y$ the pushout exists in the category of schemes. So in particular one can glue schemes along a closed point or glue a scheme to itself along a pair of closed points (although one needs to choose a subscheme structure - you would want to take the reduced induced subscheme structure probably). A reference for this (carried out via the category of locally ringed spaces) is given in this paper of Schwede (Corollary 3.9).

In general though one can the pushout in the category of locally ringed spaces need not necessarily glue more generally be a scheme even if one pushes out along a subscheme - see for instance Example 3.3 in Schwede's paper.

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Given schemes $X,Y$ and $Z$ such that $Z$ is a closed subscheme of both $X$ and $Y$ the pushout exists in the category of schemes. So in particular one can glue schemes along a closed point or glue a scheme to itself along a pair of closed points (although one needs to choose a subscheme structure - you would want to take the reduced induced subscheme structure probably). A reference for this (carried out via the category of locally ringed spaces) is given in this paper of Schwede (Corollary 3.9).

In general though one can not necessarily glue more generally - see for instance Example 3.3 in Schwede's paper.