ag.algebraic geometry – When are there no maps from a variety of high dimension to a variety of low dimension?

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 $X = Ztimes Y to Y$ 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.