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Criterion for the irreducibility of fibre products

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$begingroup$

Suppose that $V$, $W_1$, $W_2$ are smooth, irreducible varieties and $f_i:W_irightarrow W_i$ is a surjective rational map with finite fibres. In general, the corresponding fibre product $W_1times_V W_2$ is not irreducible. For example, if $W_1$ is the curve $y^2-x=0$ and $W_1$ is the curve $z^2+x=0$, then the fibre product $W_1times_mathbbA^1W_2$ is the union of the varieties $y^2-x, z-iy$ and $y^2-x, z+iy$.

I was wondering if there is some general criterion for when such a fibre product is irreducible.

What if the maps $f_i$ are just projections?

algebraic-geometry

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edited yesterday

greg

asked yesterday

greggreg

296211

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I guess $f_i$ are maps $W_i to V$ and they are regular --- otherwise it is not clear how the fiber product could be defined.
  $endgroup$
  – Sasha
  yesterday
  

  
  
  
  

add a comment | 

0

$begingroup$

Suppose that $V$, $W_1$, $W_2$ are smooth, irreducible varieties and $f_i:W_irightarrow W_i$ is a surjective rational map with finite fibres. In general, the corresponding fibre product $W_1times_V W_2$ is not irreducible. For example, if $W_1$ is the curve $y^2-x=0$ and $W_1$ is the curve $z^2+x=0$, then the fibre product $W_1times_mathbbA^1W_2$ is the union of the varieties $y^2-x, z-iy$ and $y^2-x, z+iy$.

I was wondering if there is some general criterion for when such a fibre product is irreducible.

What if the maps $f_i$ are just projections?

algebraic-geometry

share|cite|improve this question

edited yesterday

greg

asked yesterday

greggreg

296211

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I guess $f_i$ are maps $W_i to V$ and they are regular --- otherwise it is not clear how the fiber product could be defined.
  $endgroup$
  – Sasha
  yesterday
  

  
  
  
  

add a comment | 

$begingroup$

Suppose that $V$, $W_1$, $W_2$ are smooth, irreducible varieties and $f_i:W_irightarrow W_i$ is a surjective rational map with finite fibres. In general, the corresponding fibre product $W_1times_V W_2$ is not irreducible. For example, if $W_1$ is the curve $y^2-x=0$ and $W_1$ is the curve $z^2+x=0$, then the fibre product $W_1times_mathbbA^1W_2$ is the union of the varieties $y^2-x, z-iy$ and $y^2-x, z+iy$.

I was wondering if there is some general criterion for when such a fibre product is irreducible.

What if the maps $f_i$ are just projections?

algebraic-geometry

share|cite|improve this question

edited yesterday

greg

asked yesterday

greggreg

296211

$endgroup$

share|cite|improve this question

edited yesterday

greg

asked yesterday

greggreg

296211

share|cite|improve this question

edited yesterday

greg

asked yesterday

greggreg

296211

asked yesterday

greggreg

296211

asked yesterday

greggreg

296211