Let $X,Y,Z$ be reduced algebraic varieties, and let $Y$ and $Z$ be normal. Let $f:X \to Y$ and $g:X \to Z$ two surjective projective morphisms of algebraic varieties such that the geometric fibers of $f$ and $g$ coincide. Are $f$ and $g$ the same morphism (maybe up to Is there an isomorphism $Y\cong Z$)h:Y\to Z$ such that $g=h \circ f$?
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morphisms that coincide set theoreticallydo geometric fibers determine scheme-theoretic image? |
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Let $X,Y,Z$ be irreducible reduced algebraic varieties, and let $Y$ and $Z$ be normal. Let $f:X \to Y$ and $g:X \to Z$ two projective morphisms of algebraic varieties such that the geometric fibers of $f$ and $g$ coincide. Are $f$ and $g$ the same morphism (maybe up to an isomorphism $Y\cong Z$)? |
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Let $X,Y,Z$ be irreducible algebraic varieties, and let $Y$ and $Z$ be normal. Let $f:X \to Y$ and $g:X \to Z$ two projective morphisms of algebraic varieties such that $Y$ and $Z$ coincide set-theoretically, and such that the geometric fibers of $f$ and $g$ coincideas varieties. Are $f$ and $g$ the same morphism (maybe up to an isomorphism $Y\cong Z$)? |
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