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Let $X\rightarrow Y$ be a smooth morphism of schemes and consider the blowup of the fiber product along the diagonal, $W:=Bl_{\Delta}X\times_Y X$.

a.Does there exist a collection of smooth morphisms of schemes $X_i\rightarrow Y_i$ such that

1.the schemes $W_i:=Bl_{\Delta_i}X_i\times_{Y_i} X_i$ are irreducible and

2.there exist morphisms $W_i\rightarrow W$ ,such that $\{W_i\rightarrow W\}$ is an etale cover?

[Here $\Delta_i$ denotes the diagonal of $X_i\times_{Y_i} X_i$.]

b.Same question with the additional assumption that $Y$ is smooth.

In case $Y$ is singular, the answer is negative, see Will Sawin's answer below.

Let $X\rightarrow Y$ be a smooth morphism of schemes and consider the blowup of the fiber product along the diagonal, $W:=Bl_{\Delta}X\times_Y X$.

a.Does there exist a collection of smooth morphisms of schemes $X_i\rightarrow Y_i$ such that

1.the schemes $W_i:=Bl_{\Delta_i}X_i\times_{Y_i} X_i$ are irreducible and

2.there exist morphisms $W_i\rightarrow W$ ,such that $\{W_i\rightarrow W\}$ is an etale cover?

[Here $\Delta_i$ denotes the diagonal of $X_i\times_{Y_i} X_i$.]

b.Same question with the additional assumption that $Y$ is smooth.

In case $Y$ is singular, the answer in a. is negative, see Will Sawin's answer below.

Let $X\rightarrow Y$ be a smooth morphism of schemes and consider the blowup of the fiber product along the diagonal, $W:=Bl_{\Delta}X\times_Y X$.

Does there exist a collection of smooth morphisms of schemes $X_i\rightarrow Y_i$ such that

1.the schemes $W_i:=Bl_{\Delta_i}X_i\times_{Y_i} X_i$ are irreducible and

2.there exist morphisms $W_i\rightarrow W$ ,such that $\{W_i\rightarrow W\}$ is an etale cover?

[Here $\Delta_i$ denotes the diagonal of $X_i\times_{Y_i} X_i$.]

Let $X\rightarrow Y$ be a smooth morphism of schemes and consider the blowup of the fiber product along the diagonal, $W:=Bl_{\Delta}X\times_Y X$.

a.Does there exist a collection of smooth morphisms of schemes $X_i\rightarrow Y_i$ such that

1.the schemes $W_i:=Bl_{\Delta_i}X_i\times_{Y_i} X_i$ are irreducible and

2.there exist morphisms $W_i\rightarrow W$ ,such that $\{W_i\rightarrow W\}$ is an etale cover?

[Here $\Delta_i$ denotes the diagonal of $X_i\times_{Y_i} X_i$.]

b.Same question with the additional assumption that $Y$ is smooth.

In case $Y$ is singular, the answer is negative, see Will Sawin's answer below.

Let $X\rightarrow Y$ be a smooth morphism of schemes and consider the blowup of the fiber product along the diagonal, $W:=Bl_{\Delta}X\times_Y X$.

Does there exist a collection of smooth morphisms of schemes $X_i\rightarrow Y_i$ such that

1.the schemes $W_i:=Bl_{\Delta_i}X_i\times_{Y_i} X_i$ are irreducible and

2.there exist morphisms $W_i\rightarrow W$ ,such that $\{W_i\rightarrow W\}$ is an etale cover?

[Here $\Delta_i$ denotes the diagonal of $X_i\times_{Y_i} X_i$.]

Let $X\rightarrow Y$ be a smooth morphism of schemes and consider the blowup of the fiber product along the diagonal, $W:=Bl_{\Delta}X\times_Y X$.

Does there exist a collection of smooth morphisms of schemes $X_i\rightarrow Y_i$ such that

1.the schemes $W_i:=Bl_{\Delta_i}X_i\times_{Y_i} X_i$ are irreducible and

2.there exist morphisms $W_i\rightarrow W$ ,such that $\{W_i\rightarrow W\}$ is an etale cover?

[Here $\Delta_i$ denotes the diagonal of $X_i\times_{Y_i} X_i$.]