I always found Chapter II, Proposition 7.14 very intuitionless. The statement is correct and I understand the proof but it leaves me blind. Can anyone translate its hypotheses to the standard notion for varieties? That is, if I have three varieties $X,Y,Z$ over an algebraically closed field $K$ and I blow up $X$ along the rational map $h : X \to Y$ to obtain $\tilde X$, what is the condition that this result gives me for a morphism $f : Z \to X$ to extend uniquely to a morphism $g : Z \to \tilde X$? I can draw the diagram (involving $f$,$g$ and the graph of $h$), but I am still left clueless.
2026-03-26 08:03:23.1774512203
Universal property of blowing up in Hartshorne
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