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Let $X$ be a smooth divisor of degree $(1,1,2)$ in $\mathbb{P}^{1} \times \mathbb{P}^{1} \times \mathbb{P}^{2}$.

According to the list of Fano $3$-folds of Ivskovskikh-Prokhorov, this is a curve blow-up of $Y = \mathbb{P}^{1} \times \mathbb{P}^{2}$, in two different ways.

Question(s): How to describe (both) $ C \subset \mathbb{P}^{1} \times \mathbb{P}^{2}$ such that $Bl_{C}(Y)= X$?

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    $\begingroup$ Just consider each projection $\text{pr}_{1,3}$, resp. $\text{pr}_{2,3}$, from $X$ to $\mathbb{P}^1\times \mathbb{P}^2$. For homogeneous coordinates $(s,t)$ on the "forgotten" factor $\mathbb{P}^1$, $X$ is $\text{Zero}(as+bt)$ for pullbacks via projection of sections $a$ and $b$ of $\mathcal{O}(1,2)$. The curve $C$ is the common zero locus of $a$ and $b$ in $\mathbb{P}^1\times \mathbb{P}^2$. If $a$ and $b$ are general, this curve is a genus $3$ curve that projects isomorphically to a plane quartic in $\mathbb{P}^2$ and whose projection to $\mathbb{P}^1$ has degree $4$. $\endgroup$ May 3, 2018 at 13:50
  • $\begingroup$ Thanks for the answer. I don't quite understand, there are a set of "forgotten" coordinates for both projections, so when you write $as+bt$ (which I interoperate as $p_{1,2}^{*}(a)s + p_{2,3}^{*}(b)t$), which $[s:t]$ are you referring to? $\endgroup$
    – Nick L
    May 3, 2018 at 16:39
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    $\begingroup$ For the projection $\text{pr}_{1,3}$, resp. for the projection $\text{pr}_{2,3}$, the coordinates $(s,t)$ form a basis for the vector space of global sections on $\mathbb{P}^1\times \mathbb{P}^1\times \mathbb{P}^2$ of the invertible sheaf $\text{pr}_2^*\mathcal{O}(1),$ resp. the invertible sheaf $\text{pr}_1^*\mathcal{O}(1)$. $\endgroup$ May 3, 2018 at 16:47

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