Morphisms from projective space to smaller projective space are constant

A nonconstant morphism ${\mathbb{P}}^m\to {\mathbb{P}}^n$ with $m>n$ would have a positive-dimensional fiber of a morphism by the fiber dimension theorem. A target hyperplane avoiding its image point meets the positive-dimensional projective image, so its pullback is a nonempty divisor in ${\mathbb{P}}^m$ disjoint from that fibre. Every positive-dimensional projective subvariety meets every hypersurface: on its affine cone, a homogeneous equation cuts a positive-dimensional zero set containing a nonzero cone point. This contradiction proves constancy.