# 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.