Blowing Up Schemes
To pursue blow-up in its greatest generality, let X be a scheme, and let be a coherent sheaf of ideals on X. The blow-up of X with respect to is a scheme along with a morphism
such that is an invertible sheaf, characterized by this universal property: for any morphism f: Y → X such that is an invertible sheaf, f factors uniquely through π.
Notice that
has this property; this is how the blow-up is constructed. Here Proj is the Proj construction on graded sheaves of commutative rings.
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