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I've been constructing genus 2 curve fibrations by starting with a (3,2) curve in $\mathbb{P}^1 \times \mathbb{P^1}$ and then promoting its scalar coefficients to sections of line bundles over some compact complex non-singular base variety $B$. So can I construct a $\mathbb{P}^1 \times \mathbb{P^1}$-bundle by just taking the product of two $\mathbb{P}^1$ bundles? Another way to to this would be to take a smooth quadric bundle, but my equations are easily written as a bi-homogeneous polynomial so I'd like to do things in a $\mathbb{P}^1 \times \mathbb{P^1}$ bundle. Thanks.

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    $\begingroup$ Certainly fibre products of two $\mathbb{P}^1$-bundles will work. Thats it? $\endgroup$ Aug 10, 2011 at 20:44
  • $\begingroup$ That's it (just wanted to be sure thanks). $\endgroup$
    – DZN
    Aug 10, 2011 at 23:42
  • $\begingroup$ I guess $\mathbb{P}^1$-bundles over a $\mathbb{P}^1$-bundle should work also. $\endgroup$
    – DZN
    Aug 17, 2011 at 4:48

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