Let $F: X\to \mathbb{P}^n$ be a morphism from an affine variety to projective space (over some algebraically closed field of characteristic zero). Can we always find an open immersion $\iota: X\hookrightarrow\tilde{X}$ to some projective variety $\tilde{X}$ such that there is a morphism $\tilde{F}:\tilde{X}\to\mathbb{P}^n$ that extends $F$, i.e. $F=\tilde{F}\circ\iota$? The intuition says that this should be possible using blow-ups but I haven't found a suitable reference.


closed as off-topic by abx, LSpice, user44191, Dima Pasechnik, Stanley Yao Xiao Jul 14 at 19:24

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    $\begingroup$ Consider the graph $\Gamma \subset X\times \mathbb{P}^n$ of $F$ — it is isomorphic to $X$. Embed $X$ into some projective space $\mathbb{P}$, and take for $\tilde{X} $ the closure of $\Gamma $ in $\mathbb{P}\times \mathbb{P}^n$. $\endgroup$ – abx Jul 8 at 13:59