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Let $X$ be a variety over a base field $k$ of dimension $n$. Can there be non constant morphisms $P^m \to X$?

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Your question is answered negatively here (assuming $m>n$).

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  • $\begingroup$ That's great, thank you! Google did not turn up that thread at all :( $\endgroup$
    – Asvin
    Jul 16, 2019 at 19:07

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