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It's easy to show that the only maps from $\mathbb P^{n+d} \to \mathbb P^n$ are the constant maps for $d \geq 1$. Given two smooth, projective varieties $X,Y$ of dimensions $n+d,n$ as above, are there any nice, general conditions under which the only maps between them are constant? It's not always true as the example $X = Z\times Y \to Y$ shows.

By Noether normalization, we can assume that $Y = \mathbb P^n$ so I believe we are looking for conditions on $X$ so that any $n+1$ divisors on $X$ intersect non trivially.

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  • $\begingroup$ Why do you suspect the conditions on intersection of n+1 divisors? $\endgroup$
    – Evans Gambit
    Jul 18, 2021 at 6:01
  • $\begingroup$ @EvansGambit I had in my mind the proof in OP in your linked post but I might have mixed up trivially with non trivially and n with n+1. $\endgroup$
    – Asvin
    Jul 18, 2021 at 6:13

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For what it's worth, here is a discussion on families of examples where such a property holds.

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  • $\begingroup$ Aha, it seems like my question is pretty much a duplicate. Let me close my question then and thank you for the link! $\endgroup$
    – Asvin
    Jul 18, 2021 at 6:11

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