= 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.
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