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Let $f: X \to Y$ be a morphism of smooth surfaces and $C \subset Y$ be a contractible exceptional curve on $Y$. Assume that $f^{-1}(C)$ is a contractible exceptional curve as well. Denote by $X'$ (resp. $Y'$) the blow down of $f^{-1}(C)$ (resp. $C$). Does there exist a morphism $g:X' \to Y'$ such that $X$ is the fiber product $Y \times_{Y'} X'$ and the natural morphism from $Y \times_{Y'} X'$ to $Y$ is the same as $f$? Here the morphism from $Y$ to $Y'$ is the natural blow-up morphism

Let $f: X \to Y$ be a morphism of smooth surfaces and $C \subset Y$ be a contractible exceptional curve on $Y$. Assume that $f^{-1}(C)$ is a contractible exceptional curve as well. Denote by $X'$ (resp. $Y'$) the blow down of $f^{-1}(C)$ (resp. $C$). Does there exist a morphism $g:X' \to Y'$ such that $X$ is the fiber product $Y \times_{Y'} X'$ and the natural morphism from $Y \times_{Y'} X'$ to $Y$ is the same as $f$? Here the morphism from $Y$ to $Y'$ is the natural blow-up morphism.

Let $f: X \to Y$ be a morphism of smooth surfaces and $C \subset Y$ be a contractible exceptional curve on $Y$. Assume that $f^{-1}(C)$ is a contractible exceptional curve as well. Denote by $X'$ (resp. $Y'$) the blow down of $f^{-1}(C)$ (resp. $C$). Does there exist a morphism $g:X' \to Y'$ such that $X$ is the fiber product $Y \times_{Y'} X'$ and the natural morphism from $Y \times_{Y'} X'$ to $Y$ is the same as $f$? Here the morphism from $Y$ to $Y'$ is the natural blow-up morphism.

Let $f: X \to Y$ be a morphism of smooth surfaces and $C \subset Y$ be a contractible exceptional curve on $Y$. Assume that $f^{-1}(C)$ is a contractible exceptional curve as well. Denote by $X'$ (resp. $Y'$) the blow down of $f^{-1}(C)$ (resp. $C$). Does there exist a morphism $g:X' \to Y'$ such that $X$ is the fiber product $Y \times_{Y'} X'$ and the natural morphism from $Y \times_{Y'} X'$ to $Y$ is the same as $f$? Here the morphism from $Y$ to $Y'$ is the natural blow-up morphism

Let $f: X \to Y$ be a morphism of smooth surfaces and $C \subset Y$ be a contractible exceptional curve on $Y$. Assume that $f^{-1}(C)$ is a contractible exceptional curve as well. Denote by $X'$ (resp. $Y'$) the blow down of $f^{-1}(C)$ (resp. $C$). Does there exist a morphism $g:X' \to Y'$ such that $X$ is the fiber product $Y \times_{Y'} X'$ and the natural morphism from $Y \times_{Y'} X'$ to $Y$ is the same as $f$? Here the morphism from $Y$ to $Y'$ is the natural blow-up morphism

Let $f: X \to Y$ be a morphism of smooth surfaces and $C \subset Y$ be a contractible exceptional curve on $Y$. Assume that $f^{-1}(C)$ is a contractible exceptional curve as well. Denote by $X'$ (resp. $Y'$) the blow down of $f^{-1}(C)$ (resp. $C$). Does there exist a morphism $g:X' \to Y'$ such that $X$ is the fiber product $Y \times_{Y'} X'$ and the natural morphism from $Y \times_{Y'} X'$ to $Y$ is the same as $f$? Here the morphism from $Y$ to $Y'$ is the natural blow-up morphism.