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Let $S$ be a base scheme, $f \colon X \to S$ a finitely presented morphism, and $Y \to S$ an affine morphism. Suppose that $f$ is faithfully flat and separated with connected reduced geometric fibers. Also, suppose that $f$ is proper over a (topologically) dense subscheme $U$ of $S$. Is it true that every morphism $g \colon X \to Y$ over $S$ is constant (i.e., factors through a section $S \to Y$)?

Let $S$ be a base scheme and $f \colon X \to S$ and $Y \to S$ finitely presented morphisms. Suppose that $g$ is affine and $f$ is faithfully flat and separated with connected reduced geometric fibers. Also, suppose that $f$ is proper over a (topologically) dense subscheme $U$ of $S$. Is it true that every morphism $g \colon X \to Y$ over $S$ is constant (i.e., factors through a section $S \to Y$)?

Let $S$ be a base scheme, $f : X \to S$ a finitely presented morphism, and $Y \to S$ an affine morphism. Suppose that $f$ is faithfully flat and separated with connected reduced geometric fibers. Also, suppose that $f$ is proper over a (topologically) dense subset $U$ of $S$. Is it true that every morphism $g : X \to Y$ over $S$ is constant (i.e., factors through a section $S \to Y$)?

Let $S$ be a base scheme, $f \colon X \to S$ a finitely presented morphism, and $Y \to S$ an affine morphism. Suppose that $f$ is faithfully flat and separated with connected reduced geometric fibers. Also, suppose that $f$ is proper over a (topologically) dense subscheme $U$ of $S$. Is it true that every morphism $g \colon X \to Y$ over $S$ is constant (i.e., factors through a section $S \to Y$)?

Let $S$ be a base scheme, and $f : X \to S$ and $Y \to S$ finitely presented morphisms. Suppose that $f$ is faithfully flat and proper with connected reduced geometric fibers, and that $Y$ is affine over $S$. Is it true that every morphism $g : X \to Y$ over $S$ is constant (i.e., factors through a section $S \to Y$)?

Let $S$ be a base scheme,   $f : X \to S$ a finitely presented morphism, and $Y \to S$ an affine morphism. Suppose that $f$ is faithfully flat and separated with connected reduced geometric fibers. Also, suppose that $f$ is proper over a (topologically) dense subset $U$ of $S$. Is it true that every morphism $g : X \to Y$ over $S$ is constant (i.e., factors through a section $S \to Y$)?