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.