= Pole divisor avoiding a fiber
Suppose a nonconstant <morphism of varieties> $f:X\to Z$ has fiber $Y$ over $z_0$, where $X$ is normal, connected and projective. Choose an affine neighborhood $U$ of $z_0$ and a regular function on $U$ nonconstant on $f(X)\cap U$. Its pullback is a nonconstant rational function on $X$, so its pole <Weil divisor> is nonzero. A pole-free rational function on a normal variety is regular, and global regular functions on a connected <projective variety> are constant. The pole divisor avoids $f^{-1}(U)$, hence avoids $Y$. No projectivity assumption on $Z$ is needed.
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